Unimodular Hausdorff and Minkowski dimensions

This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley's lemma and Frostman's lemma are presented. These lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff measure and unimodular dimension function are also introduced. This toolbox is used to connect the unimodular dimensions to other notions such as growth rate (various further connections will also be considered in future papers of the authors). It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. This work is structured in two papers, with the present paper being the first.


Introduction
Infinite discrete random structures are ubiquitous: random graphs, branching processes, point processes, graphs or zeros of discrete random walks, discrete or continuum percolation, to name a few. The large scale and macroscopic properties of such spaces have been thoroughly discussed in the literature. In particular, various notions of dimension have been proposed; e.g., the mass dimension and the discrete (Hausdorff) dimension defined by Barlow and Taylor [8] for subsets of Z d .
The main novelty of the present paper is the definition of new notions of dimension for a class of discrete structures that, heuristically, enjoy a form of statistical homogeneity. The mathematical framework proposed to handle such structures is that of unimodular (random) discrete spaces, where unimodularity is defined here by a version of the mass transport principle. This framework unifies unimodular random graphs and networks, stationary point processes (under their Palm version) and point-stationary point processes. It does not require more than a metric; for instance, no edges or no underlying Euclidean spaces are needed. The statistical homogeneity of such spaces has been used to define localized versions of global notions such as intensity. The novelty of the present paper is the use of this homogeneity to define the notions of unimodular Minkowski and Hausdorff dimensions, which are inspired by the analogous classical notions. The definitions are obtained naturally from the classical setting by replacing the infinite sums pertaining to infinite coverings by the expectation of certain random variables at the origin (which is a distinguished point), and also by considering large balls instead of small balls. These definitions are local but capture macroscopic (large scale) properties of the space.
The definitions are complemented by a toolbox for the analysis of unimodular dimensions. Several analogues of the important results known about the classical Hausdorff and Minkowski dimensions are established, like for instance the comparison of the unimodular Minkowski and Hausdorff dimensions as well as unimodular versions of Billingsley's lemma and Frostman's lemma. These lemmas allow one to connect the dimension to the (polynomial) volume growth rate of the space, which is also called mass dimension or fractal dimension in the literature. While many ideas in this toolbox are imported from the continuum setting, their adaptation is nontrivial and there is no automatic way to import results from the continuum to the discrete setting. For some results, the statements fundamentally differ from their continuum analog; e.g., the statement of Billingsley's lemma.
These notions of dimension are complemented by further definitions which can be used for a finer study of dimension. An analogue of the Hausdorff measure is defined, which is called the unimodular Hausdorff size here. This can be used to compare sets with the same dimension. The notion of unimodular dimension function is also defined for a finer quantification of the dimension. Such notions are new for discrete spaces to the best of the authors' knowledge. Another new notion introduced in the present paper is that of regularity for unimodular spaces, which is the equality of the unimodular Minkowski and Hausdorff dimensions. Similar notions of regularity exist in the continuum setting (see e.g., the definition of fractals in [14]) and for subsets of Z d [9].
The paper also contains new mathematical results of independent interest. A weak version of Birkhoff's pointwise ergodic theorem is stated for all unimodular discrete spaces. A unimodular version of the max-flow min-cut theorem is also proved for unimodular one-ended trees, which is used in the proof of the unimodular Frostman lemma. Also, for unimodular one-ended trees, a relation between the volume growth rate and the height of the root is established as explained below.

Summary of the Main Definitions and Results
Recall that the ordinary Minkowski dimension of a compact metric space X is defined using the minimum number of balls of radius needed to cover X. Now, consider a (unimodular) discrete space D (it is useful to have in mind the example D = Z k to see how the definitions work). It is convenient to consider coverings of D by balls of equal but large radius. Of course, if D is unbounded, then an infinite number of balls is needed to cover D. So one needs another measure to assess how many balls are used in a covering. Let S ⊆ D be the set of centers of the balls in the covering. The idea pursued in this paper is that if D is unimodular, then the intensity of S is a measure of the average number of points of S per points of D (S should be equivariant for the intensity to be defined, as discussed later). This gives rise to the definition of the unimodular Minkowski dimension naturally.
The idea behind the definition of the unimodular Hausdorff dimension is similar.
Recall that the α-dimensional Hausdorff content of a compact metric space X is defined by considering the infimum of i R α i , where the R i 's are the radii of a sequence of balls that cover X. Also, it is convenient to enforce an upper bound on the radii. Now, consider a unimodular discrete space D and a covering of D by balls which may have different radii. Let R(v) be the radius of the ball centered at v. It is convenient to consider a lower bound on the radii, say R(·) ≥ 1. Again, if D is unbounded, then v R(v) α is always infinite. The idea is to leverage the unimodularity of D and to consider the average of the values R(·) α per point as a replacement of the sum. Under the unimodularity assumption, this can be defined by E [R(o) α ], where o stands for the distinguished point of D (called the origin) and where, by convention, R(o) is zero if there is no ball centered at o. This is used to define the unimodular Hausdorff dimension of D in a natural way.
The volume growth rate of the space is the polynomial growth rate of #N r (o), where N r (o) represents the closed ball of radius r centered at the origin and #N r (o) is the number of points in this ball. It is shown that the upper and lower volume growth rates of #N r (o) (i.e., limsup and liminf of log(#N r (o))/log r as r → ∞) provide upper and lower bound for the unimodular Hausdorff dimension, respectively. This is a discrete analogue of Billingsley's lemma (see e.g., [14]). A discrete analogue of the mass distribution principle is also provided, which is useful to derive upper bounds on the unimodular Hausdorff dimension. In the Euclidean case (i.e., for pointstationary point processes equipped with the Euclidean metric), it is shown that the unimodular Minkowski dimension is bounded from above by the polynomial decay rate of E [1/#N n (o)]. Weighted versions of these inequalities, where a weight is assigned to each point, are also presented. As a corollary, a weak form of Birkhoff's pointwise ergodic theorem is established for all unimodular discrete spaces. These results are very useful for calculating the unimodular dimensions in many examples. An important result is an analogue of Frostman's lemma. Roughly speaking, this lemma states that the mass distribution principle is sharp if the weights are chosen appropriately. This lemma is a powerful tool to study the unimodular Hausdorff dimension. In the Euclidean case, another proof of Frostman's lemma is provided using a version of the max-flow min-cut theorem for unimodular one-ended trees, which is of independent interest.
Depending on whether one defines the unimodular Minkowski dimension as the decay rate or the growth rate of the optimal intensity of the coverings by balls of radius r, one gets positive or negative dimensions. The present paper adopts the convention of positive dimensions for the definitions of both the unimodular Minkowski and Hausdorff dimensions, despite some mathematical arguments in favor of negative dimensions. Further discussion on the matter is provided in Subsection 8.3.

Organization of the Material
Section 2 defines unimodular discrete spaces and equivariant processes, which are needed throughout. Section 3 presents the definitions of the unimodular Minkowski and Hausdorff dimensions and the unimodular Hausdorff size. It also provides some basic properties of these unimodular dimensions as part of the toolbox for the analysis of unimodular dimensions. Various examples are discussed in Section 4. These examples are used throughout the paper. Section 5 is focused on the connections with volume growth rates and contains the statements and proofs of the unimodular Billingsey lemma and of the mass distribution principle. The unimodulat Frostman lemma is discussed in Section 7. Section 6 completes the analysis of the examples discussed in Section 4 and also discusses new examples for further illustration of the results. Section 8 discusses further topics on the matter. This includes a discussion of the connections to earlier notions of dimensions for discrete sets, in particular those proposed by Barlow and Taylor in [8,9], as well as a discussion on negative dimensions. A collection of conjectures and open problems are also listed in this section.
Throughout the paper, some easier proofs as well as some extra details (e.g., measuretheoretic requirements) are skipped for the sake of brevity. These proofs and details are nevertheless available in the arXiv version [7] of the present paper. Precise indications on where to find them in these preprints are given in the text.

Unimodular Discrete Spaces
The main objective of this section is the definition of unimodular discrete spaces as a common generalization of unimodular graphs, Palm probabilities and point-stationary point processes. If the reader is familiar with unimodular random graphs, he or she can restrict attention to the case of unimodular graphs and jump to Subsection 2.5 at first reading.

Notation and Definitions
The following notation will be used throughout. The set of nonnegative real (resp. integer) numbers is denoted by R ≥0 (resp. Z ≥0 ). The minimum and maximum binary operators are denoted by ∧ and ∨ respectively. The number of elements in a set A is denoted by #A, which is a number in [0, ∞]. If P (x) is a property about x, the indicator 1 {P (x)} is equal to 1 if P (x) is true and 0 otherwise.
Discrete metric spaces (discussed in details in Subsection 2.2) are denoted by D, D , etc. Graphs are an important class of discrete metric spaces. So the symbols and notations are mostly borrowed from graph theory.
For r > 0, N r (v) :=N r (D, v) denotes the closed r-neighborhood of v ∈ D; i.e., the set of points of D with distance less than or equal to r from v. An exception is made for r = 0 Definition 2.1. Let µ be a probability measure on a measurable space X and w : X → R ≥0 be a measurable function. Assume 0 < c := X w(x)dµ(x) < ∞. By biasing µ by w we mean the probability measure ν on X defined by ν(A) := 1 c A w(x)dµ(x).

The Space of Pointed Discrete Spaces
Throughout the paper, the metric on any metric space is denoted by d, except when explicitly mentioned. In this paper, it is always assumed that the discrete metric spaces under study are boundedly finite; i.e., every set included in a ball of finite radius in Unimodular Hausdorff and Minkowski dimensions  Note that the expectations may be finite or infinite.
When there is no ambiguity, the term g[D, o, v] is also denoted by g D (o, v) or simply g(o, v). The sum in the left (respectively right) side of (2.1) is called the outgoing mass from o (respectively incoming mass into o) and is denoted by g + (o) (respectively g − (o)). The same notation can be used for the terms in (2.2). So (2.1) and (2.2) can be summarized by These equations are called the mass transport principle in the literature. The reader will find further discussion on the mass transport principle and unimodularity in [2] and the examples therein.
As a basic example, every finite metric space D, equipped with a random root o ∈ D chosen uniformly, is unimodular. Also, the lattices of the Euclidean space rooted at 0; e.g., [Z k , 0] and [δZ k , 0], are unimodular. In addition, unimodularity is preserved under weak convergence, as observed in [13] for unimodular graphs.
The following two examples show that unimodular discrete spaces unify unimodular graphs and point-stationary point processes. Most of the examples in this work are of these types. Example 2.6 (Unimodular Random Graphs). In the case of random rooted graphs and networks, the concept of unimodularity in Definition 2.5 coincides with that of [2] (see also Remark A.5 of [7] regarding the topologies). Therefore, unimodular random graphs and networks are special cases of unimodular (marked) discrete spaces. Example 2.7 (Point-Stationary Point Processes). Point-stationarity is defined for point processes Φ in R k such that 0 ∈ Φ a.s. (see e.g., [39]). This definition is equivalent to (2.1), except that g is required to be invariant under translations only (and not under all isometries). This implies that [Φ, 0] is unimodular. In addition, by considering the mark m(x, y) := y − x on pairs of points of Φ, point-stationarity of Φ will be equivalent to the unimodularity of [Φ, 0; m] (see also Remark A.5 of [7] regarding the topologies). Note also that Φ can be recovered from [Φ, 0; m]. For example, if Φ is a stationary point process in R k (i.e., its distribution is invariant under all translations), with finite intensity (i.e., a finite expected number of points in the unit cube), then the Palm version of Φ is a point-stationary point process, where the latter is heuristically obtained by conditioning Φ to contain the origin (see e.g., Section 13 of [18] for the precise definition). Also, if (X n ) n∈Z is a stochastic process in R k with stationary increments such that X 0 = 0, then the graph of this random walk is a point-stationary point process in R k+1 . The image of this random walk is also pointstationary provided that it is discrete (i.e., the random walk is transient) and X i = X j a.s. for every i = j. See Subsection 4.3.

Equivariant Process on a Unimodular Discrete Space
In many cases in this paper, an unmarked unimodular discrete space [D, o] is given and various ways of assigning marks to D are considered. Intuitively, an equivariant process on D is an assignment of (random) marks to D such that the new marked space is unimodular. Formally, it is a unimodular marked discrete space [D , o ; m] such that the space [D , o ], obtained by forgetting the marks, has the same distribution as [D, o].
In this paper, it is more convenient to work with a disintegrated form of this heuristic, defined below. It can be proved that the two notions are equivalent, but the proof is skipped for brevity (this claim is similar to invariant disintegration for group actions). The easy part of the claim is Lemma 2.12 below. For the other direction, see Proposition B.1 of [7].
In the following, the mark space Ξ is fixed as in Subsection 2.2. Definition 2.8. Let D be a deterministic discrete space which is boundedly-finite. A marking of D is a function from D × D to Ξ; i.e., an element of Ξ D×D . A random marking of D is a random element of Ξ D×D . Definition 2.9. An equivariant process Z with values in Ξ is a map that assigns to every deterministic discrete space D a random marking Z D of D satisfying the following properties: (i) Z is compatible with isometries in the sense that for every isometry ρ : D 1 → D 2 , the random marking Z D1 • ρ −1 of D 2 has the same distribution as Z D2 .
(ii) For every measurable subset A ⊆ D * , the following function on D * is measurable: In addition, given a unimodular discrete space [D, o], such a map is also called an equivariant process on D. In this case, one can also let Z (·) be undefined for a class of discrete spaces, as long as it is defined for almost all realizations of D. It is important that extra randomness be allowed here.
Convention 2.10. If D is clear from the context, Z D (·) is also denoted by Z(·) for simplicity.
Note that in the above definition, D is deterministic and is not an equivalence class of discrete spaces. However, for an equivariant process on [D, o], one can define [D, o; Z D ] as a random pointed marked discrete space with distribution Q (on D * ), defined by where P D is the distribution of Z D (for every D) and µ is the distribution of [D, o] (note that only the distribution of Z D is important here and it doesn't matter which probability space is used for Z D ). It can be seen that Q(A) is indeed well defined and is a probability measure on D * . As mentioned before, the probabilities and expectations to be used for Z D and [D, o; Z D ] will be denoted by the same symbols P and E; e. The proof is straightforward and skipped for brevity. The converse of this claim also holds (see [7]). It is important here to assume that the distribution of Z D does not depend on the origin (as in Definition 2.9).
Remark 2.13. One can easily extend the definition of equivariant processes to allow the base space to be marked. Therefore, for point-stationary point processes, one can replace condition (i) by invariance under translations only (see Example 2.7). In particular, every stationary stochastic process on Z k defines an equivariant process on Z k . The above lemma is a generalization of similar results in [5] and [2].

Notes and Bibliographical Comments
The mass transport principle was introduced in [31]. The concept of unimodular graphs was first defined for deterministic transitive graphs in [12] and generalized to random rooted graphs and networks in [2]. Unimodular graphs have many analogies and connections to (Palm versions of) stationary point processes and point-stationary point processes, as discussed in Example 9.5 of [2] and also in [5] and [35]. As already explained, the framework of unimodular discrete spaces introduced in this section can be regarded as a common generalization of these concepts.
Special cases of the notion of equivariant processes have been considered in the literature. The first formulation in Subsection 2.5 is considered in [2] for unimodular graphs. Factors of IID [40] are special cases of equivariant processes where the marks of the points are obtained from i.i.d. marks (Example 2.11) in an equivariant way. Covariant subsets and covariant partitions of unimodular graphs are defined similarly in [5], but no extra randomness is allowed therein. In the case of stationary (marked) point processes, the first formulation of Subsection 2.5 is used in the literature. However, the authors believe that the general formulation of Definition 2.9 is new even in those special cases. udim M (Φ) = decay 1 r r 0 p(s)ds ≤ 1 ∧ decay (p(r)) , udim M (Φ) = decay 1 r r 0 p(s)ds = 1 ∧ decay (p(r)) .
Proof. Let r > 0 and ϕ be a discrete subset of R. Let U r be a random number in [0, r) chosen uniformly. For each n ∈ Z, put a ball of radius r centered at the largest element of ϕ ∩ [nr + U r , (n + 1)r + U r ). Denote this random r-covering of ϕ by R ϕ . One can see that R is equivariant under translations (see Remark 2.13). This implies that R is an equivariant covering (verifying Condition (ii) of Definition I. 2.9 is skipped here). One has Now, since R is a 3-bounded covering, Lemma 3.9 implies the two left-hand-side equalities. For all β < decay (p(r)), one has p(r) < r −β for large enough r. So, if in addition, β < 1, then q(r) < cr −β for some constant c, so that decay (q(r)) ≥ β. Therefore decay (q(r)) ≥ 1 ∧ decay (p(r)). Now, the final equality in the claim is deduced from q(r) ≥ p(r). Similarly, if decay (p(r)) < 1, one can deduce decay (q(r)) ≤ decay (p(r)). Also, q(r) ≥ 1 r 1 0 p(s)ds, and hence decay (q(r)) ≤ 1. This implies the first inequality and completes the proof.

The Unimodular Hausdorff Dimension
The definition of the unimodular Hausdorff dimension is based on coverings of the discrete space by balls of possibly different radii. Such a covering can be represented by an assignment of marks to the points, where the mark of a point v represents the radius of the ball centered at v. As mentioned earlier, it is convenient to assume that the radii are at least 1 (in fact, this condition is technically necessary in what follows). Also, by convention, if there is no ball centered at v, the mark of v is defined to be 0. In relation with this convention, the following notation is used for all discrete spaces D and points v ∈ D: In words, N r (v) is the closed ball of radius r centered at v, except when r = 0. (v) will also be denoted by N R (v). Also, for 0 ≤ α < ∞ and 1 ≤ M < ∞, let where the infimum is over all equivariant coverings R such that almost surely, ∀v ∈ D : , and, by convention, 0 0 := 0. Note that H α M (D) is a non-decreasing function of both α and M .
In the ergodic case, E [R(o) α ] can be interpreted as the average of R(·) α over the vertices. Also, P [R(o) > 0] (which is used for defining the unimodular Minkowski dimension) can be interpreted as the number of balls per point. Ergodicity is however a special case, and there is no need to assume it in what follows; for more on the matter, see Example 3.19 and the discussion after it.  with the convention that sup ∅ = 0.
The key point of assuming equivariance in the above definition is that by Lemma   Proof. Let R be an arbitrary equivariant covering. For all discrete spaces D and u, v ∈ D, let g D (u, v) be 1 if d(u, v) ≤ R D (u) and 0 otherwise. One has g + (u) = #N R (u) and g − (u) ≥ 1 a.s. (since R is a covering). By the assumption and the mass transport Since R is arbitrary, one gets H α 1 (D) ≥ 1 c > 0, and hence, udim H (D) ≤ α. shown below that udim M (D) = udim H (D) = 1. For n ∈ N, the equivariant n-covering of Example 3.3 makes sense for D and is uniformly bounded. One has P [R(0) > 0] = 1 2 (n −1 + n −2 ). This implies that udim M (D) = decay 1 2 (n −1 + n −2 ) = 1. Also, for α < 1, one has E [R(o) α ] = 1 2 (n α−1 + n α−2 ) → 0 as n → ∞. This implies that H α 1 (D) = 0 for all α < 1 and hence udim H (D) ≥ 1. On the other hand, for any equivariant covering S, one has Remark 3.20. The result of this example might seem counterintuitive at first glance as the union of a filled square and a segment is two dimensional. The number of balls of radius required to cover the square dominates the number of balls required to cover the segment, but in Example 3.19, the situation is reversed: a larger fraction of points is needed to cover Z than Z 2 . This is a consequence of considering large balls and also counting the number of balls per point. See also Subsection 8.3. In fact, the following example justifies more clearly why Example 3.19 is one dimensional: Let G n be the union of a n × n square grid (regarded as a graph) and a path of length n 2 sharing a vertex with the grid. To cover G n by balls of radius r, a fraction of order 1/r of the vertices of G n are needed (as r is fixed and n → ∞). So it is not counterintuitive to say that G n is one dimensional asymptotically. Indeed, G n tends to the random graph of Example 3.19 in the local weak convergence [2] as n → ∞ (if one chooses the root of G n randomly and uniformly). instance, in some examples, it is convenient to condition on having infinite cardinality (which is common, e.g., in branching processes). However, in general, it doesn't seem easier to define the dimension of samples separately in a way that is compatible with the definitions of this paper. In the future work [6], the notion of sample dimension is defined by combining the definitions in this paper with either ergodic decomposition or conditional expectation. In this work, the reader may focus mainly on the ergodic case, but it should be noted that the definitions and results do not require ergodicity. Proof. The first inequality holds by the definition. For the second one, the definition of λ r (3.1) implies that for every α ≥ 0 and r ≥ 1,

Comparison of Hausdorff and Minkowski Dimensions
This readily implies that H α 1 (D) ≤ r α λ r for every r ≥ 1. So, if α < decay (λ r ), one gets H α 1 (D) = 0, and hence, udim H (D) ≥ α. This implies the claim. follows, the equality udim M (D) = udim H (D) will be referred to as regularity for the unimodular discrete space D, regarded as a fractal object.

The Unimodular Hausdorff Size
Consider the setting of Subsection 3.3. For 0 ≤ α < ∞, let where H α M (D) is defined in (3.3). Note that the limit exists because of monotonicity.
Proof. (i). If R is an equivariant covering, them M R is also an equivariant covering and satisfies ∀v ∈ D : M R(v) ∈ {0} ∪ [M, ∞) a.s. (ii). The claim is implied by part (i). (iii). If R is an equivariant covering such that ∀v ∈ D :   Finally, assume D is finite with probability q. For all deterministic discrete spaces D, let R D be one of the above coverings depending on whether D is finite or infinite. It satisfies P [R(o) > 0] = E [1/#D] + p(1 − q). Since p is arbitrary, the claim is proved. Proof. Let S n be the covering in Example 3.3 scaled by factor δ. One has E S n (o) k = (nδ) k /(2n + 1) k . This easily implies that H k ∞ (δZ k ) ≤ (δ/2) k . On the other hand, the proof of Lemma 3.17 shows that H k ∞ (δZ k ) ≥ cδ k , where c is any constant such that r k ≥ c#N r (0) for large enough r. It follows that H k ∞ (δZ k ) ≥ (δ/2) k , and the claim is proved.  , o] be a unimodular discrete space and d be an equivariant metric. If d ≤ cd + a a.s., with c and a constants, then the dimension of (D, d ) is larger than or equal to that of (D, d). Moreover, for every α ≥ 0,

The Effect of a Change of Metric
Proof. The claim is implied by the fact that the ball N cr+a ((D, d ), v) contains the ball N r ((D, d), v) and is left to the reader.
As a corollary, if 1 c d − a ≤ d ≤ cd + a a.s., then (D, d ) has the same unimodular dimensions as (D, d). Also, cD has the same dimension as D and M α (cD) = c −α M α (D).
For instance, this result can be applied to Cayley graphs, which are an important class of unimodular graphs [2]. It follows that the unimodular dimensions of a Cayley graph do not depend on the generating set. In fact, it will be proved in Subsection 6.6 that these dimensions are equal to the polynomial growth degree of H. Example 3.32. Let [G, o] be a unimodular graph. Examples of equivariant metrics on G are the graph-distance metric corresponding to an equivariant spanning subgraph (e.g., the drainage network model of Subsection 4.5 below) and metrics generated by equivariant edge lengths. More precisely, if l is an equivariant process which assigns a positive weight to the edges of every deterministic graph, then one can let d (u, v) be the minimum weight of the paths that connect u to v. If d is a metric for almost every realization of G and is boundedly-finite a.s., then it is an equivariant metric.   (ii) If ρ is the intensity of S in D, then for every α ≥ 0, the α-dim H-size of S D satisfies

Dimension of Subspaces
Theorem 3.34 is proved below by using the fact that every covering of the larger set induces a covering of the subset by deleting some balls and then re-centering and enlarging the remaining balls. This matches the analogous idea in the continuum setting. The apparently surprising direction of the inequalities is due to the definition of dimension which implies that having less balls means having larger or equal dimension. For more on the matter, see the discussion on negative dimension in Subsection 8.3.  is guaranteed if S D is a r-covering of D for some constant r. In other words, roughly speaking, the unimodular dimensions are quasi-isometry invariant (see e.g., [28]) and do not depend on the fine details of the discrete space.
Proof of Theorem 3.34. The first claim of (i) is implied by (ii) and Lemma 3.26, and hence, is skipped. Let R be an arbitrary equivariant r-covering of D. For every v ∈ R, let τ (v) be an element picked uniformly at random in N r (v) ∩ S D , which is defined only where the equality is by the mass transport principle. This gives ρλ 2r (S D ) ≤ λ r (D), which implies the claims regarding the Minkowski dimension.
implies that there exists a sequence R n of equivariant coverings of S D such that R n (·) ∈ {0} ∪ [n, ∞) for all n = 1, 2, . . . and E [R n (o) α |o ∈ S D ] → H α ∞ (S D ). One may extend R n to be defined on D by letting R n (v) := 0 for v ∈ D \S D . Let > 0 be arbitrary and B n ⊆ D be the union of N (1+ )Rn (v) for all v ∈ D. Define R n (u) := (1 + )R n (u) for u ∈ B n and R n (u) := 1/ for u ∈ B n . It is clear that R n is an equivariant covering of D. Also, Since the radii of the balls in R n are at least n, one gets that B n includes the n- Note that the radii of the balls in R n are at least n ∧ (1/ ). Therefore, one obtains Conversely, let R n be a sequence of equivariant coverings of D for n = 1, 2, It can be seen that R n is an equivariant covering of S D . One has where the equality is by the mass transport principle. It follows that and the claim is proved.

Covering By Arbitrary Sets
According to Remark 3.35, it is more natural to redefine the Hausdorff size by considering coverings by finite subsets which are not necessarily balls (as in the continuum setting). A technical challenge is to define such coverings in an equivariant way. This will be done at the end of this subsection using the notion of equivariant processes of Subsection 2.5. Once an equivariant covering C is defined (which is an equivariant collection of finite subsets), one can define the average diameter of sets U ∈ C per point The same idea is used to redefine H α M (D) as follows: where the infimum is over all equivariant coverings C. Here, taking the maximum with M is similar to the condition that the subsets have diameter at least 2M (note that a ball of radius M might have diameter strictly less than 2M ). Finally, define the modified unimodular Hausdorff size M α (D) similarly to (3.6). Remark 3.35 shows an advantage of this definition. Also,the reader can verify that This implies that the notion of unimodular Hausdorff dimension is not changed by this modification. One can also obtain a similar equivalent form of the unimodular Minkowski dimension. This is done by redefining λ r by considering equivariant coverings by sets of diameter at most 2r. The details are left to the reader. A similar idea will be used in Subsection 4.1.2 to calculate the Minkowski dimension of one-ended trees. Finally, here is the promised representation of the above coverings as equivariant processes (it should be noted that it is not always possible to number the subsets in an equivariant way and the collection should be necessarily unordered). To show the idea, and uniformly. Note that multiple marks are assigned to every point and the covering can be reconstructed from the marks. With this idea, let the mark space Ξ be the set of discrete subsets of R 2 (regard every discrete set as a counting measure and equip Ξ with a metrization of the vague topology). This mark space can be used to represent equivariant coverings by equivariant processes (for having a complete mark space, one can extend Ξ to the set of discrete multi-sets in R 2 ).

Notes and Bibliographical Comments
Several definitions and basic results of this section have analogues in the continuum setting. A list of such analogies is given below. Note however that there is no systematic way of translating the results in the continuum setting to that of unimodular discrete spaces. In particular, inequalities are most often, but not always, in the other direction. The comparison of the unimodular Minkowski and Hausdorff dimensions (Theorem 3.22) is analogous to the similar comparison in the continuum setting (see e.g., (1.2.3) of [14]), but in the reverse direction. Theorem 3.31, regarding changing the metric, is analogous to the fact that the ordinary Minkowski and Hausdorff dimensions are not increased by applying a Lipschitz function. Theorem 3.34 regarding the dimension of subsets is analogous to the fact that the ordinary dimensions do not increase by passing to subsets. Note however that equality holds in Theorem 3.34 for the unimodular Hausdorff dimension (and also for the unimodular Minkowski dimension in most usual examples), in contrast to the continuum setting.
For point processes (Example 2.7), one can redefine the unimodular Hausdorff dimension by using dyadic cubes instead of balls. This changes the value of the Hausdorff size up to a constant factor, and hence, the value of Hausdorff dimension is not changed. Since dyadic cubes are nested, this simplifies some of the arguments. This approach will be used in Subsection 7.3.

Examples
This section presents a set of examples of unimodular discrete spaces together with discussions about their dimensions. Recall that the tools for bounding the dimensions are summarized in Remarks 3.5 and 3.18. As mentioned in Remark 3.18, bounding the Hausdorff dimension from above usually requires the unimodular mass transport principle or the unimodular Billingsley lemma, which will be stated in Section 5. So the upper bounds for some of the following examples are completed later in Subsection 6.1.

General Unimodular Trees
In this subsection, general results are presented regarding the dimension of unimodular trees with the graph-distance metric. Specific instances are presented later in the section. It turns out that the number of ends of the tree plays a key role (an end in a tree is an equivalence class of simple paths in the tree, where two such paths are equivalent if their symmetric difference is finite).
It is well known that the number of ends in a unimodular tree belongs to {0, 1, 2, ∞} [2]. Unimodular trees without end are finite, and hence, are zero dimensional (Example 3.16). The only thing to mention is that there exists an algorithm to construct an optimal n-covering for such trees. This algorithm is similar to the algorithm for one-ended trees, discussed below, and is skipped for brevity. The case of infinitely many ends will be studied in Subsection 6.2. The remaining two cases are discussed below.

Unimodular Two-Ended Trees
If T is a tree with two ends, then there is a unique bi-infinite path in T called its trunk.
Moreover, each connected component of the complement of the trunk is finite.
Proof. For all two-ended trees T , let S T be the trunk of T . Then, S is an equivariant subset. Therefore, Theorem 3.34 implies that udim H (T ) = udim H (S T ). Since the trunk is isometric to Z as a metric space, Example 3.16 implies that udim H (T ) = 1. In addition, Remark 3.35 and Proposition 3.
The claim concerning the unimodular Minkowski dimension is implied by Corollary 5.10 of the next section, which shows that any unimodular infinite graph satisfies udim M (G) ≥ 1 (this theorem will not be used throughout).

Unimodular One-Ended Trees
Unimodular one-ended trees arise naturally in many examples (see [2]). In particular, the (local weak) limit of many interesting sequences of finite trees/graphs are one-ended ( [3,2]). In terms of unimodular dimensions, it will be shown that unimodular one-ended trees are the richest class of unimodular trees. First, the following notation is borrowed from [5]. Every one-ended tree T can be regarded as a family tree as follows. For every vertex v ∈ T , there is a unique infinite simple path starting from v. Denote by F (v) the next vertex in this path and call it the parent of v. By deleting F (v), the connected component containing v is finite. This set is denoted by D(v) and its elements are called the descendants of v. The maximum distance of v to its descendants will be called the height of v and be denoted by h(v).    The proof of Theorem 4.2 is based on a recursive construction of an optimal covering by cones, defined below, rather than balls. It is shown below that considering cones instead of balls does not change the Minkowski dimension. An optimal ball-covering is also discussed in [7].
The cone with height n at v ∈ T is defined by C n (v) := N n (v) ∩ D(v); i.e., the first n generations of the descendants of v, including v itself. Let λ n be the infimum intensity of equivariant coverings by cones of height n. The claim is that λ 2n ≤ λ n ≤ λ n .  To prove (4.4), note that any covering by cones of height n is also a covering by balls of radius n. This implies that λ n ≤ λ n . Also, if S is a covering by balls of radius n, then {F n (v) : v ∈ S} is a covering by cones of height 2n. By the mass transport principle (2.2), one can show that the intensity of the latter is not greater than the intensity of S. This implies that λ 2n ≤ λ n . So (4.4) is proved. Proof. Let A be any equivariant covering of T by cones of height n. Consider a realization (T ; A) of [T ; A]. Let v be a vertex such that h(v) = n. Since A is a covering by cones of height n, A should have at least one vertex in D(v) (to see this, consider the farthest leaf from v in D(v)). Now, for all such vertices v, delete the vertices in A ∩ D(v) from A and then add v to A. Let A 1 be the subset of T obtained by doing this operation for all vertices v of height n. So A 1 is also a covering of T by cones of height n. Now, remove all vertices {v : h(v) = n} and their descendants from T to obtain a new one-ended tree. Consider the same procedure for the remaining tree and its intersection with A. Inductively, one obtains a sequence of subsets A = A 0 , A 1 , . . . of T such that, for each i, A i is a covering of T by cones of height n which agrees with S T on the set of vertices that are removed from the tree up to step i. By letting [T ; A] be random, the above induction gives a sequence of equivariant subsets A = A 0 , A 1 , . . . on T . It can be seen that the intensity of A 1 is at most that of A (this can be verified by the mass transport principle (2.1)) and more generally, the hence, S is an optimal covering by cones of height n.  Let v ∈ T be an arbitrary vertex. Let k be the unique integer such that (k − 1)n − 1 < h(v) ≤ kn − 1. Let j be the first nonnegative integer such that h(F j (v)) ≥ kn − 1 and let w := F j (v). One has 0 ≤ j ≤ n − 1. By considering the longest path in D(w) from w to the leaves, one finds z ∈ D(w) such that h(z) mod n = −1 and 0 ≤ d(w, z) ≤ n − 1. Therefore where the last inequality can be verified by the mass transport principle (2.1)). This implies the second inequality in (4.6).
To prove the first inequality in (4.6), let S be the optimal covering by cones of height n given by the algorithm of Lemma 4.3. Send unit mass from each vertex v ∈ S to the first vertex in v, F (v), . . . , F n (v) which belongs to B n+1 (if there is any). So the outgoing mass from v is at most 1 {v∈S} . In the next paragraph, it is proved that the incoming mass to each w ∈ B n+1 is at least 1. This in turn (by the mass transport principle) implies that , which proves the first inequality in (4.6).
The final step consists in proving that the incoming mass to each w ∈ B n+1 is at least 1. If h(w) = n, then w ∈ S and the claim is proved. So assume h(w) > n. By considering the longest path from w in D(w), one can find a vertex z such that w = F n+1 (z) and h(z) = h(w) − (n + 1). This implies that no vertex in {F (z), . . . , F n (z)} is in B n+1 . So to prove the claim, it suffices to show that at least one of these vertices or w itself lies in S. Note that in the algorithm in Lemma 4.3, at each step, the height of w decreases by a value at least 1 and at most n + 1 until w is removed from the tree. So in the last step before w is removed, the height of w is in {0, 1, . . . , n}. This is possible only if in the same step of the algorithm, an element of {F (z), . . . , F n (z), w} is added to S. This implies the claim and the lemma is proved. Now, the tools needed to prove the main results are all available. The result then follows by the mass transport principle. This monotonicity implies that These inequalities conclude the proof of (4.1) and (4.2). It remains to prove (4.3). The second inequality follows from (4.1) and the fact for each i. One may assume the sequence is such that n i ≥ 2 i for each i. Now, for each k ∈ N, consider the following covering of T : . By arguments similar to Lemma 4.4, it can be seen that R k is indeed a (ball-)covering. It It is easy to see that the maximum of the function ( .7) is proved and the proof is completed.

Instances of Unimodular Trees
This subsection discusses the dimension of some explicit unimodular trees. More examples are given in Subsection 4.5, in Section 6, and also in the ongoing work [6] (e.g., uniform spanning forests).

The Canopy Tree
The canopy tree C k with offspring cardinality k [1] is constructed as follows. Its vertex set is partitioned in levels L 0 , L 1 , . . .. Each vertex in level n is connected to k vertices in level n − 1 (if n = 0) and one vertex (its parent) in level n + 1. Let o be a random vertex of C k such that P [o ∈ L n ] is proportional to k −n . Then, [C k , o] is a unimodular random tree.
Below, three types of metrics are considered on C k . First, consider the graph-distance metric. Given n ∈ N, let S : Second, for each n, let the length of each edge between L n and L n+1 be a n , where a > 1 is constant. Let d 1 be the resulting metric on C k . Given r > 0, let S 1 be the set of vertices having distance at least r/a to L 0 (under d 1 ). One can show that S 1 is an r-covering of (C k , d 1 ) and decay (P [o ∈ S 1 ]) = log k/ log a. Therefore, udim M (C k , d 1 ) ≥ log k/ log a. On the other hand, one can see that the ball of radius a n centered at o (under d 1 ) has cardinality of order k n . One can then use Lemma 3.17 to show that Third, replace a n by n! in the second case and let d 2 be the resulting metric. Then, the cardinality of the ball of radius r centered at o has order less than r α for every α > 0. One can use Lemma 3.17 again to show that udim

The Generalized Canopy Tree
This example generalizes the canopy tree of Construct a graph T on Φ as follows: For each n, connect each x ∈ Φ n to its closest point (or closest point on its right) in Φ n+1 . Note that T is a forest by definition. However, the next lemma shows that [T , o] is a unimodular tree.  . This is skipped for brevity (see [7]). The main ingredients are using stationarity of Φ under horizontal translations and the fact that Φ n − o n is point-stationary (the proof is similar to that of the formula for the Palm version of the superposition of stationary point processes, e.g., in [47].) To prove (ii), note that T can be realized as an equivariant process on Φ (see Definition 2.9 and Remark 2.13). Therefore, by Lemma 2.12 and Theorem 3.31, it is enough to prove that T is connected a.s. Nevertheless, the same lemma implies that the connected component T of T containing o is a unimodular tree. Since it is one-ended, Theorem 3.9 of [5] implies that the foils T ∩ Φ i are infinite a.s. By noting that the edges do not cross (as segments in the plane), one obtains that T ∩ Φ i should be the whole Φ i ; hence, T = T . Therefore, T is connected a.s. and the claim is proved.
Proposition 4.7. The sequence (p n ) n can be chosen such that where T is endowed with the graph-distance metric. Moreover, for any 0 ≤ α ≤ β ≤ γ ≤ ∞, the sequence (p n ) n can be chosen such that Proof. T is a one-ended tree (see Subsection 4.1.2). Assume the sequence (p n ) n is non-increasing. So the construction implies that there is no leaf of the tree in Φ n for all n > 0. Therefore, for all n ≥ 0, the height of every vertex in Φ n is precisely n. So by letting q n := i≥n p i , Theorem 4.2 implies that For simplicity, assume 0 < α and γ < ∞ (the other cases can be treated similarly). Define n 0 , n 1 , . . . recursively as follows. Let n 0 := 0. Given that n i is defined, let n i+1 be large enough such that the line connecting points (n i , n −β i ) and (n i+1 , n −β i+1 ) intersects the graph of the function x −α and has slope larger than −n −γ . Now, let q ni := n −β i for each i and define q n linearly in the interval [n i , n i+1 ]. Let p n := q n − q n+1 . It can be seen that p n is non-increasing, decay (q n ) ≤ α, decay (q n ) = β and decay (p n ) ≥ γ.

Unimodular Eternal Galton-Watson Trees
Eternal Galton-Watson (EGW) trees are defined in [5]. Unimodular EGW trees (in the nontrivial case) can be characterized as unimodular one-ended trees in which the descendants of the root constitute a Galton-Watson tree. Also, unimodularity implies that the latter Galton-Watson tree is necessarily critical (use the mass transport principle when sending a unit mass from each vertex to its parent ). Here, the trivial case that each vertex has exactly one offspring is excluded (where the corresponding EGW tree is a bi-infinite path). In particular, the Poisson skeleton tree [3] is an eternal Galton-Watson tree.
Recall that the offspring distribution of a Galton-Watson tree is the probability where p n is the probability that the root has n offsprings.

Examples Associated with Random Walks
Let µ be a probability measure on R k . Consider the (double-sided) simple random walk (S n ) n∈Z in R k starting from S 0 := 0 such that and the jumps S n − S n−1 are i.i.d. with distribution µ. In this subsection, unimodular discrete spaces are constructed based on the image and the zero set of this random walk and their dimensions are studied in some special cases. The graph of the simple random walk will be studied in Subsection 6.4.

The Image of the Simple Random Walk
Assume the random walk is transient; i.e., it visits every given ball only finitely many times. It follows that the image Φ = {S n } n∈Z is a random discrete subset of R k . If no point of R k is visited more than once (e.g., when S n is in the positive cone a.s.), then it can be seen that Φ is a point-stationary point process, hence, [Φ, 0] is a unimodular discrete space. Hence, [Φ, 0] is a unimodular discrete space. In the general case, by similar arguments, one should bias the distribution of [Φ, 0] by the inverse of the multiplicity of the origin; i.e., by 1/#{n : S n = 0}, to obtain a unimodular discrete space. This claim can be proved by direct verification of the mass transport principle.
Below, the focus is on the case where the jumps are real-valued and strictly positive.
In this case, Φ is actually a point stationary renewal process [24].
So the claims regarding the Minkowski dimension are direct consequences of Proposition 3.13 and do not require the i.i.d. assumption. The proofs of the last two claims, will be given in Subsection 6.1.
The image of the nearest-neighbor simple random walk in Z k will be studied in [6]. It will be shown that it has dimension 2 when k ≥ 2. Furthermore, a doubling property will be proved in this case.
As another example, if [T , o] is any unimodular tree such that the simple random walk on T is transient a.s., then the image of the (two sided) simple random walk on T is another unimodular tree (after biasing by the inverse of the multiplicity of the root). The new tree is two-ended a.s., and hence, is 1-dimensional by Theorem 4.1. Proof. Represent Ψ uniquely as Ψ := {S n : n ∈ Z} such that S 0 := 0 and S n < S n+1 for each n. Then, (S n ) n is another simple random walk and Ψ is its image. The distribution of the jump S 1 is explicitly computed in the classical literature on random walks (using the reflection principle). In particular, there exist c 1 , c 2 > 0 such that c 1 r − 1 2 < P [S 1 > r] < c 2 r − 1 2 for every r ≥ 1. Therefore, the claim is implied by part (iv) of Proposition 4.9

Zeros of the Simple Random Walk
(recall that this part of Proposition 4.9 will be proved later).

A Subspace with Larger Minkowski Dimension
Let Φ ⊆ R be an arbitrary point-stationary point process. Let S 1 be the first point of Φ on the right of the origin. Assume β := decay (P [S 1 > r]) exists with β < 1. Then Proposition 4.9 gives that udim M (Φ) = β.
Let α < β < 1. Consider the intervals defined by consecutive points of Φ. In each such interval, say (a, b), add (b − a) α − 1 points so as to split the interval into (b − a) α equal parts. Let Φ denote the resulting point process (with the points of Φ and the additional points). The assumption α < β implies that E [S α 1 ] < ∞. Now, by biasing the distribution of Φ by S α 1 and changing the origin to a point of Φ ∩ [0, S 1 ) chosen uniformly at random, one obtains a point-stationary point process Ψ (see Theorem 5 in [35] and also the examples in [2]), (it is not a renewal process). The distribution of Ψ is determined by the following equation, where h is any measurable nonnegative function: Proposition 4.11. Let Φ and Ψ be as above. Then, Φ has the same distribution as an equivariant subspace of Ψ (conditioned on having the root) and Note that Theorem 3.34 implies that udim H (Φ) = udim H (Ψ). Therefore, the proposition implies udim M (Ψ) < udim H (Ψ).
Proof. Let A be the set of newly-added points in Ψ, which can be defined by adding marks from the beginning and is an equivariant subset of Ψ. By (4.8), one can verify that Ψ \ A conditioned on 0 ∈ A has the same distribution as Φ (see also Proposition 6 in [35]). Also, by letting c := E [ S α 1 ], (4.8) gives Now, by the assumption decay (P [S 1 > r]) = β and integration by parts, it is straightforward to deduce that decay (P [Ψ ∩ (0, r) = 0]) = (β − α)/(1 − α). Therefore, Proposition 3.13 gives the claim.
Remark 4.12. The fact that Ψ has a smaller Minkowski dimension than Φ means that the tail of the distribution of the jumps (or inter-arrivals) of Ψ is heavier than that of the inter-arrivals of Φ. This may look surprising as the inter-arrival times of Ψ are obtained by subdividing those of Φ into smaller sub-intervals. The explanation of this apparent contradiction is of the same nature as that of Feller's paradox (Section I.4 of [24]). It comes from the renormalization of size-biased sampling: the typical inter-arrival of Ψ has more chance to be found in a larger inter-arrival of Φ, and this length-biasing dominates the effect of the subdivision.

A Drainage Network Model
Practical observations show that large river basins have a fractal structure. For example, [30] discovered a power law relating the area and the height of river basins. There are various ways to model river basins and their fractal properties in the literature.
In particular, [45] formalizes and proves a power law with exponent 3/2 for a specific model called Howard's model. Below, the simpler model of [43] is studied. One can ask similar questions for Howard's model or other drainage network models.
Connect each (x, y) in the even lattice {(x, y) ∈ Z 2 : x + y mod 2 = 0} to either (x − 1, y − 1) or (x + 1, y − 1) with equal probability in an i.i.d. manner to obtain a directed graph T . Note that the downward path starting at a given vertex is the rotated graph of a simple random walk. It is known that T is connected and is a one-ended tree (see e.g., [45]). Also, by Lemma 2.12, [T , 0] is unimodular.
Note that by considering the Euclidean metric on T , the Hausdorff dimension of T is 2. In the following, the graph-distance metric is considered on T . Proof (first part). Here, it will be proved that udim M (T ) = 3 2 . The rest of the proof is postponed to Subsection 6.1. The idea is to use Theorem 4.2. Following [45], there are two backward paths (going upward) in the odd lattice that surround the descendants D(o) of the origin. These two paths have exactly the same distribution as (rotated) graphs of independent simple random walks starting at (−1, 0) and (1, 0), respectively, until they hit for the first time. In this setting, h(o) is exactly the hitting time of these random walks. So classical results on random walks imply that P [h(o) ≥ n] is bounded between two constant multiples of n − 1 2 for all n. So Theorem 4.2 implies that udim M (T ) = 3

Self Similar Unimodular Discrete Spaces
This section provides a class of examples of unimodular discrete spaces obtained by discretizing self-similar sets. Let l ≥ 1 and f 1 , . . . , f l be similitudes of R k with similarity ratios r 1 , . . . , r l respectively (i.e., ∀x, y ∈ R k : |f i (x) − f i (y)| = r i |x − y|). For every n ≥ 0 and every string σ = (j 1 , . . . , j n ) ∈ {1, . . . , l} n , let f σ := f j1 · · · f jn . Also let |σ| := n. Fix a point o ∈ R k (one can similarly start with a finite subset of R k instead of a single point). Let K 0 := {o} and K n+1 := j f j (K n ) for each n ≥ 0. Equivalently, Recall that if r i < 1 for all i, then by contraction arguments, K n converges in the Hausdorff metric to the attractor of f 1 , . . . , f l (see e.g., Section 2.1 of [14]). The attractor is the unique compact set K ⊆ R k such that K = i f i (K then the Minkowski and Hausdorff dimensions of K are equal to the similarity dimension, which is the unique α ≥ 0 such that r α i = 1. The following is the main result of this section. It introduces a discrete analogue of self-similar sets by scaling the sets K n and taking local weak limits.    Here, it is easy to see that the random set Ψ n := 3 n (K n − o n ) ⊆ Z converges weakly to the random set Ψ ⊆ Z defined as follows: Ψ := ∪ n T n , where T n is defined by letting T 0 := {0, ±1} and T n+1 := T n ∪ (T n ± 2 × 3 n ), where the sign is chosen i.i.d., each sign with probability 1/2. Note that each T n has the same distribution as Ψ n , but the sequence T n is nested. In addition, since o n is chosen uniformly, Ψ n and Ψ are point-stationary point processes, and hence [Ψ, 0] is unimodular (a deterministic discrete Cantor set exists in the literature which is not unimodular).
Theorem 4.14 implies that udim M (Ψ) = udim H (Ψ) = log 2/log 3.  C n chosen uniformly and Φ n := 3 n (C n − x n ). It can be seen that Φ n tends weakly to a random discrete subset Φ of the triangular lattice which is almost surely a bi-infinite path (note that the cycle disappears in the limit). It can be seen that Φ can be obtained by Theorem 4.14. In this paper, Φ is called the unimodular discrete Koch snowflake. Also, Theorem 4.14 implies that udim M (Φ) = udim H (Φ) = log 4/log 3.
In addition, Φ can be constructed explicitly as Φ := ∪ n T n , where T n is a random finite path in the triangular lattice with distinguished end points A n and B n defined inductively as follows: Let T 1 := {A 1 , B 1 }, where A 1 is the origin and B 1 is a neighbor of the origin in the triangular lattice chosen uniformly at random. For each n ≥ 1, given (T n , A n , B n ), let (T n+1 , A n+1 , B n+1 ) be obtained by attaching to T n three isometric copies of itself as shown in Figure 1. There are 4 ways to attach the copies and one of them should be chosen at random with equal probability (the copies should be attached to T n relative to the position of A n and B n ). It can be seen that no points overlap.

Remark 4.19.
If the r i 's are not all equal, the guess is that there is no scaling of the sequence [K n , o n ] that converges to a nontrivial unimodular discrete space (which is not a single point). This has been verified by the authors in the case o ∈ V . In this case, by letting a n be the distance of o n to its closest point in K n , it is shown that for any > 0, P [a n /(r) n < ] → 1 2 and P a n /(r) n > 1 → 1 2 , wherer is the geometric mean of r 1 , . . . , r l . This implies the claim (note that the counting measure matters for convergence; e.g., {0, 1 n } does not converge to {0}).
To prove Theorem 4.14, it is useful to consider the following nested version of the sets One has w n (v) ≤ w n+1 (v). Note that in the case o ∈ V , w n (·) = 1 and the arguments are much simpler. The reader can assume this at first reading.
In the following, for x ∈ R k , B r (x) represents the closed ball of radius r centered at x in R k .  Proof. (i). Assume f σ1(o) = · · · = f σ l (o) and |σ j | = n for each j ≤ l. Let D be a fixed number such that V intersects B D (o). Now, the sets f σj (V ) for 1 ≤ j ≤ l are disjoint and intersect a common ball of radius Dr n . Moreover, each of them contains a ball of radius ar n and each is contained in a ball of radius br n (for some fixed a, b > 0). Therefore, Lemma 2.2.5 of [14] implies that l ≤ ( D+2b a ) k =: C. This implies that w n (·) ≤ C a.s., hence w(·) ≤ C a.s.  Proof of Theorem 4.14. Convergence is proved in Lemma 4.20. The rest of the proof is base on the construction of a sequence of equivariant coverings ofK. In this proof, with an abuse of notation, the dimension ofK means the dimension of the unimodular space obtained by biasing the distribution ofK by 1/w(o) (see Lemma 4.20). Let D > diam(K) be given, where K is the attractor of f 1 , . . . , f l . Let m > 0 be large enough so that diam(K m ) < D. Note that each element inK can be written as f −1 δn f σ (o) for some n and some string σ of length n. Let γ m be a string of length m chosen uniformly at random and independently of other variables. For an arbitrary n and a string σ of length n, let Note that U σ ⊆K is always a scaling of K m with ratio r −m and z σ ∈ U σ . Now, define the following covering ofK: It can be seen that R m gives an equivariant covering. Also, note that R m Therefore, by considering the biasing that makesK unimodular, one gets (4.10) Since the balls in the covering have radius Dr −m , one gets udim M (K) ≥ α.
On the other hand, by (4.10) and monotone convergence, one finds that  Consider the sets V σj := f −1 δn+m f σj (V ) which are disjoint (since σ j 's have the same length). Note that if > diam(V ∪ {o}) is fixed, then the -neighborhood of V contains K m . Therefore, all V σj 's intersect a common ball of radius D + r −m ≤ (1 + )D. Moreover, each of them contains a ball of radius ar −m ≥ arD and is contained in a ball of radius br −m ≤ bD (for some a, b > 0 not depending on D). Therefore, Lemma 2.2.5 of [14] implies that k ≤ ( (1+ )+2b ar ) k . Therefore, (4.11) implies that #B D (o) ∩K ≤ CD α , a.s. Therefore, Lemma 3.17 implies that udim H (K) ≤ α. Moreover, the proof of the lemma shows that M α (K) < ∞. This completes the proof.

Remark 4.21.
Motivated by Examples 4.17 and 4.18, it can be seen that every unimodular self similar discrete space can be constructed by successively attaching copies of a set to itself. This is expressed in the following algorithm. Proposition 4.8 is inspired by the dimension of the Brownian continuum random tree (see [29] or Theorem 5.5 of [20]), which is the scaling limit of Galton-Watson trees Unimodular Hausdorff and Minkowski dimensionŝ K 0 := {o}; Let g 0 be the identity map; Choose i.i.d. random numbers i 1 , i 2 , . . . uniformly in {1, . . . , l}; for n = 1, 2, . . . do letK n consist of l isometric copies ofK n−1 as followŝ

Notes and Bibliographical Comments
conditioned to be large. The zero set of the simple random walk (Proposition 4.10) is analogous to the zero set of Brownian motion. Self-similar unimodular discrete spaces are inspired by continuum self-similar sets (see e.g., Section 2.1 of [14]) as discussed in Subsection 4.6.

The Unimodular Mass Distribution Principle and Billingsley Lemma
Let D be a discrete space and o ∈ D.  [8], fractal dimension, or growth degree); volume growth will be used in the present paper since it is common in the context of graphs and discrete groups.
If the upper and lower volume growth rates are equal, the common value is called the volume growth rate of D. Note that for v ∈ D, one has N r (o) ⊆ N r+c (v) and N r (v) ⊆ N r+c (o), where c := d(o, v). This implies that growth (#N r (o)) and growth (#N r (o)) do not depend on the choice of the point o.
In various situations in this paper, some weight in R ≥0 will be assigned to each point of D. In these cases, it is natural to redefine the volume growth rate by considering the weights; i.e., by replacing #N r (o) with the sum of the weights of the points in N r (o).
This will be formalized below using the notion of equivariant processes. Recall that an equivariant process should be defined for all discrete spaces D. However, if a random pointed discrete space [D, o] is considered, it is enough to define weights in almost every realization (see Subsection 2.5 for more on the matter). Also, given D, the weights are allowed to be random.

Unimodular Billingsley Lemma
The main result of this subsection is Theorem 5.6. It is based on Lemmas 5.3 and 5.4 below. Lemma 5.3 is a stronger version of the mass distribution principle (Theorem 5.2).        N r (o))) ≥ α a.s., then udim H (D) ≥ α.
Proof. The proofs of the first two parts are very similar. The second part is proved first. (ii). Let β, γ and κ be such that γ < β < κ < α. Fix n ∈ N. Let S = S D be the equivariant subset obtained by selecting each point v ∈ D with probability 1 ∧ (n −β w(v)) (the selection variables are assumed to be conditionally independent given [D, o; w]).
(i). Only a small change is needed in the above proof. For n ≥ r 0 , let R n (v) = n if either v ∈ S D or N n (v) ∩ S D = ∅, and let R n (v) = 0 otherwise. Note that R n is a covering by balls of equal radii. By the same computations and the assumption M ≤ r 0 , one gets which is of order n −β for large n. This implies that udim M (D) ≥ β. Since β is arbitrary, one gets udim M (D) ≥ α and the claim is proved. (iii). Let β < α. It will be proved below that under the assumption of (iii), there is a sequence r 1 , r 2 , . . . such that E exp −r −β n w(N rn (o)) → 0. If so, by a slight modification of the proof of part (ii), one can find a sequence of equivariant coverings R n such that E R n (o) β < ∞ and (iii) is proved. Let > 0 be arbitrary. By the assumption, there is δ > 0 and r ≥ 1 such that Note that for fixed and δ as above, r can be arbitrarily large. Now, choose r large enough for the right hand side to be at most 2 . This shows that E exp −r −β w(N r (o)) can be arbitrarily small and the claim is proved.
(iv). As before, let R n (v) = n if either v ∈ S D or N n (v) ∩ S D = ∅, and let R n (v) = 0 otherwise. The calculations in the proof of part (ii) show that Now, the assumption implies the claim.

Remark 5.5.
The assumption in part (iii) of Lemma 5.4 is equivalent to the condition that there exists a sequence r n → ∞ such that the family of random variables r α n /w(N rn (o)) is tight. Also, from the proof of the lemma, one can see that this assumption is equivalent to Proof. The first inequality is implied by part (ii) of Lemma 5.4. For the second inequality, assume that growth (w(N r (o))) < α with positive probability. On this event, one has w(N r (o)) ≤ r α for large r; i.e., lim sup r w(N r (o))/r α ≤ 1. Now, Lemma 5.3 implies that udim H (D) ≤ α. This proves the second inequality. The last claim follows because growth (X n ) ≤ growth (E [X n ]) for any monotone sequence of nonnegative random variables X n (see Lemma C.3 of [7]).  N r (o))) a.s.
In fact, without the assumption of this corollary, an inequality similar to (5.5) is valid for the sample Hausdorff dimension of D, which will be studied in [6]. Analogously, for many natural fractals in the continuum setting, there is a natural mass measure that can be used in Billingsley's lemma. These assumptions are also necessary as shown below.
For example, assume Φ is a point-stationary point process in R (see Example 2.7).
For v ∈ Φ, let w(v) be the sum of the distances of v to its next and previous points in Φ. This equivariant weight function satisfies w(N r (v)) ≥ 2r for all r, and hence growth (w (N r (o))) ≥ 1. But udim H (Φ) can be strictly less than 1 as shown in Subsec-

Bounds for Point Processes
Example 2.7, explains that for point processes containing the origin, unimodularity is, roughly speaking, equivalent to point-stationarity. To study the dimension of such processes, the following covering is used in the next results. Let ϕ be a discrete subset of R k equipped with the l ∞ metric and r ≥ 1. Let C := C r := [0, r) k , U := U r be a point chosen uniformly at random in −C, and consider the partition {C + U + z : z ∈ rZ k } of R k by cubes. Then, for each z ∈ rZ k , choose a random element in (C + U + z) ∩ ϕ independently (if the intersection is nonempty). The distribution of this random element should depend on the set (C + U + z) ∩ ϕ in a translation-invariant way (e.g., choose with the uniform distribution or choose the least point in the lexicographic order). Let R =R ϕ assign the value r to the selected points and zero to the other points of ϕ. Then, R is an equivariant covering. Also, each point is covered at most 3 k times. So R is 3 k -bounded (Definition 3.8). where U r is a uniformly at random point in −C r independent of Φ and w.
Proof. By Theorem 3.31, one may assume the metric on Φ is the l ∞ metric without loss of generality. Given any r > 0, consider the equivariant covering R described above, but when choosing a random element of (C r + U r + z) ∩ ϕ, choose point v with probability w ϕ (v)/w ϕ (C r + U r + z) (conditioned on w ϕ ). One gets P [0 ∈ R] = E [w(0)/w(C r + U r )]. As mentioned above, R is equivariant and uniformly bounded (for all r > 0). So Lemma 3.9 implies both equalities in the claim. The inequalities are implied by the facts that w(C r + U r ) ≤ w(N r (0)) and which is implied by the Cauchy-Schwartz inequality.
Example 5.14. The right-most inequalities in the above theorem can be strict. For example, let T > 0 be a random number and let Φ := 1 T Z. Then #N r (0) ∼ 1 + T /r. So, decay (1/#N r (0)) = 1, but it might be the case that E [#N r (0)] = ∞. For an ergodic example, let 1 ≤ T i ∈ Z be i.i.d. with finite mean but infinite variance (for i ∈ Z). In each interval [i, i + 1], put T i − 1 equidistant points and let Φ 0 be the union of these points together with Z. Bias the distribution of Φ 0 by T 0 (Definition 2.1) and then translate Φ 0 by moving a random point in Φ 0 ∩ [0, 1) to the origin. Let Φ be the resulting point process. It can be seen that Φ is unimodular and point-stationary. Proof. One may assume the metric on Φ is the l ∞ metric without loss of generality. Let C := [0, 1) k and U be a random point in −C chosen uniformly. For all discrete subsets ϕ ⊆ R k and v ∈ ϕ, let C(v) be the cube containing v of the form C + U + z (for z ∈ Z k ) and w ϕ (v) := 1/#(ϕ ∩ C(v)). Now, w is an equivariant weight function. The construction readily implies that w(N r (o)) ≤ (2r + 1) k . Moreover, by w ≤ 1, one has E [w(0)] < ∞. Therefore, the unimodular Billingsley lemma (Theorem 5.6) implies that udim H (Φ) ≤ k. Proposition 5.16. If Ψ is a stationary point process in R k with finite intensity and Ψ 0 is its Palm version, then udim M (Ψ 0 ) = udim H (Ψ 0 ) = k. Moreover, the modified unimodular Hausdorff size of Ψ 0 , defined in Section 3.8, satisfies M k (Ψ 0 ) = 2 k ρ(Ψ), where ρ(Ψ) is the intensity of Ψ.
Notice that if Ψ 0 ⊆ Z k , then the claim is directly implied by Theorem 3.34. The general case is treated below.
Proof. For the first claim, by Proposition 5.15 and Theorem 3.22, it is enough to prove that udim M (Ψ 0 ) ≥ k. Let Ψ be a shifted square lattice independent of Ψ (i.e., Ψ = Z k +U , where U ∈ [0, 1) k is chosen uniformly, independently of Ψ). Let Ψ := Ψ ∪ Ψ . Since Ψ is a superposition of two independent stationary point processes, it is a stationary point process itself. By letting p := ρ(Ψ)/(ρ(Ψ) + 1), the Palm version Ψ 0 of Ψ is obtained by the superposition of Ψ 0 and an independent stationary lattice with probability p (heads), and the superposition of Z k and Ψ with probability 1 − p (tails). So part (i) of Lemma 5.4 implies that udim M (Ψ 0 ) ≥ k. Note that Ψ 0 has two natural equivariant subsets which, after conditioning to contain the origin, have the same distributions as Ψ 0 and Z k respectively. Therefore, one can use Theorem 3.34 to deduce that udim M (Ψ 0 ) ≥ udim M (Ψ 0 ) = k. Therefore, Proposition 5.15 implies that udim H (Ψ 0 ) = udim M (Ψ 0 ) = k.
The last claim of Proposition 5.16 suggests the following, which is verified when k = 1 in the next proposition. Proof. Denote Φ as Φ = {S n : n ∈ Z} such that S 0 = 0 and S n < S n+1 for each n. Then, the sequence T n := S n+1 − S n is stationary under shifting the indices (see Example 2.7).
The assumption that Φ is not the Palm version of a stationary point process is equivalent to E [S 1 ] = ∞ (see [18] or Proposition 6 of [35]). Indeed, if E [S 1 ] < ∞, then one could bias the probability measure by S 1 (Definition 2.1) and then shift the whole process by −U , where U ∈ [0, S 1 ] is chosen uniformly and independently.

Connections to Birkhoff's Pointwise Ergodic Theorem
The following corollary of the unimodular Billingsley lemma is of independent interest. Note that the statement does not involve dimension.   N r (o)). On the other side, note that amenability is not assumed in this Theorem, which is a general requirement in the study of ergodic theorems. However, it will be proved in [6] that, roughly speaking, nonamenability implies growth (w 2 (N r (o))) = ∞, which makes the claim of Theorem 5.19 trivial in this case. In this case, using exponential gauge functions seems more interesting.
Problem 5.21. Is it true that for every unimodular discrete space [D, o], the growth rates growth (w (N r (o))) and growth (w (N r (o))) do not depend on w as long as 0 < E [w(o)] < ∞?

Notes and Bibliographical Comments
As already mentioned, the unimodular mass distribution principle and the unimodular Billingsley lemma have analogues in the continuum setting (see e.g., [14]) and are named accordingly. Note however that there is no direct or systematic reduction to these continuum results. For instance, in the continuum setting, one should assume that the space under study is a subset of the Euclidean space, or more generally, satisfies the bounded subcover property (see e.g., [14]). Theorem 5.6 does not require such assumptions. Note also that the term growth (w(N r (o))) in Theorem 5.6 does not depend on the origin in contrast to the analogous term in the continuum version. Similar observations can be made on Theorem 5.2.

Examples Continued
This section presents further examples for illustrating the results of the previous section.

Remaining Proofs from Section 4
The By the explicit construction of EGW trees in [5], Y n is a critical Galton-Watson tree up to generation n. Also, for 0 ≤ i < n, Y i has the same structure up to generation i, except that the distribution of the first generation is size-biased minus one (i.e., (np n+1 ) n with the notation of Subsection 6.3.2). So the assumption of finite variance implies that the first generation in each Y i has finite mean, namely m . Now, one can inductively show that E [#Y n ] = n and E [#Y i ] = im , for 0 ≤ i < n. It follows that E [#N n (o)] ≤ (1 + m )n 2 and the claim is proved.
Proof of Proposition 4.9 (second part). In Subsection 4.3.1, it is proved that udim M (Φ) ≥ 1 ∧ decay (P [S 1 > r]). So part (iv) is implied by part (iii), which is proved below. Since Φ is a point-stationary point process in R (see Subsection 4.3.1), Proposition 5.15 implies that udim H (Φ) ≤ 1. Now, assume decay (P [S 1 > r]) < β. Then, there exists c > 0 such that P [S 1 > r] > cr −β for all r ≥ 1. This implies that there exists C < ∞ and a random number r 0 > 0 such that for all r ≥ r 0 , one has #N r (o) ≤ Cr β log log r a.s. (for the proof, see [7] or Theorem 4 of [26]). Therefore, the unimodular Billingsley lemma (Theorem 5.6) implies that udim H (Φ) ≤ β + for every > 0, which in turn implies that udim H (Φ) ≤ β. Example 6.1 (Infinite H-Size). In Proposition 4.9, assume that P [S 1 > r] = 1/ log r for large enough r. Then, part (iii) of the proposition implies that udim H (Φ) = 0. However, since Φ is infinite a.s., it has infinite 0-dim H-size (Proposition 3.28).  −1 (o)) be the number of descendants of order k of F l (o) which are not a descendant of F l−1 (o) (for l = 0, let it be just #F −k (o)). One has #N n (o) = k,l e k,l 1 {k+l≤n} . It can be seen that E [e k,l ] is equal to the probability that two independent paths of length k and l starting both at o do not collide at another point. Therefore, E [e k,l ] ≤ c(k ∧ l) − 1 2 for some c and all k, l. This implies that (in the following, c is updated at each step to a new constant without changing the notation) The above inequalities imply that E [#N n (o)] ≤ cn 3 2 for some c and all n. Therefore, the unimodular Billingsley lemma (Theorem 5.6) implies that udim H (T ) ≤ 3 2 . So the claim is proved.

General Unimodular Trees Continued
The following is a direct corollary of Theorem 4.2 and the unimodular Billingsley lemma. Since the statement does not involve dimension, it is of independent interest and believed to be new. The rest of this subsection is focused on unimodular trees with infinitely many ends. In fact, the graph-distance metric on T can be replaced by an arbitrary equivariant metric. This will be proved in [6].
The following proof uses the definitions and results of [2], but they are not recalled for brevity.
Proof of Proposition 6.4. By Corollary 8.10 of [2], [T , o] is non-amenable (this will be discussed further in [6]). So Theorem 8.9 of [2] implies that the critical probability p c of percolation on T is less than one with positive probability. In fact, it can be shown that p c < 1 a.s. (if not, condition on the event p c = 1 to get a contradiction). For any tree, p c is equal to the inverse of the branching number. So the branching number is more than one, which implies that the tree has exponential volume growth. Finally, the unimodular Billingsley lemma (Theorem 5.6) implies that udim H (T ) = ∞.
The following example shows that the Minkowski dimension can be finite. Example 6.5. Let T be the 3-regular tree. Split each edge e by adding a random number l e of new vertices and let T 0 be the resulting tree. Let v e be the middle vertex in this edge (assuming l e is always odd) and assign marks by m 0 (v e ) := l e . Assume that the random variables l e are i.i.d. If E [l e ] < ∞, then one can bias the probability measure and choose a new root to obtain a unimodular marked tree, namely [T , o; m] (see Example 9.8 of [2] or [35]). It will be shown below that udim M (T ) may be finite. Let R be an arbitrary equivariant r-covering of T . Consider the set of middle vertices A r := {v ∈ T : m(v) ≥ r}. Since these vertices have pairwise distance at least r, they belong to different balls in the covering. So, by the mass transport principle, one can show that ρ(R) ≥ ρ(A r ), where ρ(·) = P [o ∈ ·] denotes the intensity. On the other hand, let S be the equivariant subset of vertices with degree 3. Send unit mass from every point of A r to its two closest points in S. Then the mass transport principle implies that 2ρ(A r ) = 3ρ(S)P [l e ≥ r]. Hence, ρ(R) ≥ 3 2 ρ(S)P [l e ≥ r]. This gives that udim M (T ) ≤ decay (P [l e ≥ r]), which can be finite. In fact, if decay (P [l e ≥ r]) exists, Proposition 6.6 below implies that udim M (T ) = decay (P [l e ≥ r]).
The following proposition gives a lower bound on the Minkowski dimension. Proposition 6.6. Let [T , o] be a unimodular tree with infinitely many ends and without leaves. Let S be the equivariant subset of vertices of degree at least 3. For every v ∈ S, let w(v) be the sum of the distances of v to its neighbors in S.
The proof is based on the following simpler result. This will be used in Subsection 6.3.3 as well. Proof. Define w (v) := u d (u, v) α , where the sum is over the three closest neighbors of v. It is enough to assume that d is generated by equivariant edge lengths since increasing the edge lengths does not increase the dimension (by Theorem 3.31). By the same argument, it is enough to assume d (u, v) ≥ 1 for all u ∼ v. Then, it can be seen that there exists a constant c, that depends only on α, such that w (N r (v)) ≥ cr α for all v ∈ T and r ≥ 0 (see Lemma C.5 in [7]). Also, the assumption implies that E [w (o)] < ∞. So Lemma 5.4 implies that udim M (T 3 , d ) ≥ α and the claim is proved.
Proof of Proposition 6.6. For v ∈ S, let w (v) := u d(u, v) α , where the sum is over the neighbors of v in S. For v ∈ T \ S, if u 1 and u 2 are the two closest points of S to v, let g(v, u i ) := d(u i , v) α−1 and w (v) := g(v, u 1 ) + g(v, u 2 ). The assumption implies that E [w (o)] < ∞ (use the mass transport principle for g defined above). Similarly to Proposition 6.7, there exists c = c(α), such that w (N r (v)) ≥ cr α for all v ∈ T and r ≥ 0 (see Lemma C.5 in [7]) and the claim is proved.

A Unimodular Tree With No Volume Growth Rate
Recall the generalized canopy tree [T , o] from Subsection 4.2.2. Here, it is shown that growth (T ) = growth (T ) if the parameters are suitably chosen. Similarly, it provides an example where the exponential growth rate does not exist. The existence of unimodular trees without exponential growth rate is already proved in [49], but with a more difficult construction. n p n = 1, where c is constant and q 0 ≤ q 1 ≤ · · · is a sequence of integers. In this case, T is obtained by splitting the edges of the canopy tree by adding new vertices or concatenating them, depending only on the level of the edges. It can be seen that if v is a vertex in the n-th level of T , then the number of descendants of v is (p 0 + · · · + p n )/p n . It follows that growth (T ) = decay (p n ) and growth (T ) = decay (p n ). So, by choosing (p n ) n appropriately, T can have no polynomial (or exponential) volume growth rate. This proves the claim. Note also that the unimodular Billingsley lemma and Theorem 4.2 imply that udim H (T ) = growth (T ) here.

Unimodular Galton-Watson Trees
Here, it is shown that the unimodular Galton-Watson tree [2] is infinite dimensional (note that this tree differs from the Eternal Galton-Watson tree of Subsection 4.2.3 which is a directed tree). Consider an ordinary Galton-Watson tree with offspring distribution µ = (p 0 , p 1 , . . .), where µ is a probability measure on Z ≥0 . The unimodular Galton-Watson tree [T , o] has a similar construction with the difference that the offspring distribution of the origin is different from that of the other vertices: It has for distribution the size-biased versionμ = ( n m p n ) n , where m is the mean of µ (assumed to be finite). In what follows, the trivial case p 1 = 1 is excluded. If m ≤ 1, then T is finite a.s.; i.e., there is extinction a.s. Therefore, udim H (T ) = 0. So assume the supercritical case, namely m > 1. If p 0 > 0, then T is finite with positive probability. So udim H (T ) = 0 for the same reason. Nevertheless, one can condition on non-extinction as follows. Proof. The result for the Hausdorff dimension follows from the unimodular Billingsley lemma (Theorem 5.6) and the Kesten-Stigum theorem [34], which implies that lim n #N n (o)m −n exists and is positive a.s. Computing the Minkowski dimension is more difficult. By part (iv) of Lemma 5.4, it is enough to prove that E (1 − n −α ) #Nn(o) has infinite decay rate for every α ≥ 0. Denote by [ T , o] the Galton-Watson tree with the same parameters. Using the fact that #N n (o) is stochastically larger than #N n−1 ( o), one gets that it is enough to prove the last claim for [ T , o].
For simplicity, the proof is given for the case p 0 = 0 only. By this assumption, the probability of extinction is zero. The general case can be proved with similar arguments and by using the decomposition theorem of supercritical Galton-Watson trees (see e.g., Theorem 5.28 of [41]). In fact, the following proof implies the general claim by the fact that the trunk, conditioned on non-extinction, is another supercritical unimodular Galton-Watson tree. The latter can be proved similarly to the decomposition theorem. processes; see, e.g., [4]). One has f (0) = g(0) = 0, f (1) = g(1) = 1, f (1) = m > 1, g (1) = (1 + a)/a, and f is convex. Therefore, a can be chosen large enough such that f (s) ≤ g(s) for all s ∈ [0, 1]. So f (n) (s) ≤ g (n) (s) = a n s a n + (a + 1) n (1 − s) , where the last equality can be checked by induction. Therefore, f (n) (1 − n −α ) ≤ a n a n + n −α (a + 1) n .
It follows that decay f (n) (1 − n −α ) = ∞. Therefore, the above discussion gives that E (1 − n −α ) #Nn(o) has infinite decay rate and the claim is proved.

The Poisson Weighted Infinite Tree
The Poisson Weighted Infinite Tree (PWIT) is defined as follows (see e.g., [3]). It is a rooted tree [T , o] such that the degree of every vertex is infinite. Regarding T as a family tree with progenitor o, the edge lengths are as follows. For every u ∈ T , the set {d(u, v) : v is an offspring of u} is a Poisson point process on R ≥0 with intensity function x k , where k > 0 is a given integer. Moreover, for different vertices u, the corresponding Poisson point processes are jointly independent. It is known that the PWIT is unimodular (notice that although each vertex has infinite degree, the PWIT is boundedly-finite as a metric space). See for example [3] for more details.  [7]). This proves the claim.

The Graph of the Simple Random Walk
As in Subsection 4.3, consider the simple random walk (S n ) n∈Z in R k , where S 0 = 0 and the increments S n − S n−1 are i.i.d. The graph of the random walk (S n ) n∈Z is Ψ := {(n, S n ) : n ∈ Z} ⊆ R k+1 . It can be seen that Ψ is a point-stationary point process, and hence, [Ψ, 0] is unimodular (see Subsection 4.3.1).
Below, the focus is on the case k = 1 and on the following metric: d((x, y), (x , y )) := max{ |x − x |, |y − y |}. Proof. Let n ∈ N. The ball N n (0) has at most 2n 2 + 1 elements. So the mass distribution principle (Theorem 5.2) implies that udim H (Ψ) ≤ 2. For the other side, let C be the equivariant disjoint covering of Z 2 by translations of the rectangle [−n 2 , n 2 ] × [−n, n] (similar to Example 3.11). For each rectangle σ ∈ C, select the right-most point in σ ∩ Ψ and let S = S Ψ be the set of selected points. By construction, S gives an n-covering of Ψ and it can be seen that it is an equivariant covering. Let σ 0 be the rectangle containing the origin. By construction, 0 ∈ S if and only if it is either on a right-edge of σ 0 or on a horizontal edge of σ 0 and the random walk stays outside σ 0 . The first case happens with probability 1/(2n 2 + 1). By classical results concerning the hitting time of random walks, one can obtain that the probability of the second case lies between two constant multiples of n −2 . It follows that P [0 ∈ S] lies between two constant multiples of n −2 . Therefore, udim M (Ψ) ≥ 2. This proves the claim.

Other Self Similar Unimodular Spaces
In this subsection, two examples are presented which have some kind of self-similarity heuristically, but do not fit into the framework of Subsection 4.6.

Unimodular Discrete Spaces Defined by Digit Restriction
Let J ⊆ Z ≥0 . For n ≥ 0, consider the set of natural numbers with expansion (a n a n−1 . . . a 0 ) in base 2 such that a i = 0 for every i ∈ J. Similarly to the examples in Subsection 4.6, one can shift this set randomly and take a limit to obtain a unimodular discrete space.
This can be constructed in the following way as well: In particular, this provides another example of a unimodular discrete space where the (polynomial) volume growth rate does not exist.
Proof. Let n ≥ 0 be given. Cover T n by a ball of radius 2 n centered at the minimal element of T n . By the same recursive definition, one can cover T n+1 by either 1 or 2 balls of the same radius. Continuing the recursion, an equivariant 2 n -covering R n is obtained. It is straightforward to see that P [R n (o) > 0] = 2 −Jn . Since these coverings are uniformly bounded (Definition 3.8), Lemma 3.9 implies that udim M (Ψ) = d(J) and udim M (Ψ) = d(J). One has #T m = 2 Jm . (6.2) This implies that #N 2 n (o) ≤ 2 Jn+1 . One can deduce that growth (#N n (o)) ≤ d(J). So the unimodular Billingsley lemma (Theorem 5.6) gives udim H (Ψ) ≤ d(J). This proves the claim.

Randomized Discrete Cantor set
This subsection proposes a unimodular discrete analogue of the random Cantor set, recalled below. Let 0 ≤ p ≤ 1 and b > 1. The random Cantor set in R k [32] (see also [14]) is defined by Λ k (b, p) := ∩ n E n , where E n is defined by the following random algorithm: For each n ≥ 0 and each b-adic cube of edge length b −n in E n , divide it into b k smaller b-adic cubes of edge length b −n−1 . Keep each smaller b-adic cube with probability p and delete it otherwise independently from the other cubes. Let E n+1 be the union of the kept cubes. It is shown in Section 3.7 of [14] that Λ k (b, p) is empty for p ≤ b −k and otherwise, has dimension k + log b p conditioned on being non-empty. For each n ≥ 0, let K n be the set of lower left corners of the b-adic cubes forming E n . It is easy to show that K n tends to Λ k (b, p) a.s. under the Hausdorff metric.  Note that in contrast to the continuum analogue [32], for p = b −k , the set is non-empty and even infinite, though still zero dimensional. Also, for p < b −k the set is non-empty as well.
To prove the above proposition, the following construction ofK will be used. First, consider the usual nested sequence of partitions Π n of Z k by translations of the cube {0, . . . , b n − 1} k , where n ≥ 0. To make it stationary, shift each Π n randomly as follows. Let a 0 , a 1 , . . . ∈ {0, 1, . . . , b − 1} k be i.i.d. uniform numbers and let U n = n i=0 a i b i ∈ Z k . Shift the partition Π n by the vector U n to form a partition denoted by Π n . It is easy to see that Π n is a nested sequence of partitions. Lemma 6.13. Let (Π n ) n be the stationary nested sequence of partitions of Z k defined above. For each n ≥ 0 and each cube C ∈ Π n that does not contain the origin, with probability 1 − p (independently for different choices of C), mark all points in C ∩ Z k for deletion. Then, the set of the unmarked points of Z k , pointed at the origin, has the same distribution as [K,ô] defined in Proposition 6.12.
Proof of Lemma 6.13. Let Φ be the set of unmarked points in the algorithm. For n ≥ 0, let C n be the cube in Π n that contains the origin. It is proved below that C n ∩ Φ has the same distribution as b n (K n − o n ). This implies the claim.
Let A n ⊆ [0, 1] k be the set of possible outcomes of o n . One has #A n = b kn . For v ∈ A n , it is easy to see that the distribution of b n (K n − o n ), conditioned on o n = v, coincides with the distribution of C n ∩ Φ conditioned on C n = b n ([0, 1) k − v). So it remains to prove that P [o n = v] = P C n = b n ([0, 1) k − v) , which is left to the reader.
Here is another description ofK. The nested structure of n Π n defines a tree as follows. The set of vertices is n Π n . For each n ≥ 0, connect (the vertex corresponding to) every cube in Π n to the unique cube in Π n+1 that contains it. This tree is the canopy tree (Subsection 4.2.1) with offspring cardinality N := b k , except that the root (the cube {0}) is always a leaf. Now, keep each vertex with probability p and remove it with probability 1 − p in an i.i.d. manner. Let T be the connected component of the remaining graph that contains the root. Conditioned on the event that T is infinite,K corresponds to the set of leaves in the connected component of the root. Consider now the Minkowski dimension. As above, we assume m > 1 and the proofs for the other cases are similar. Let n ≥ 0 be given. By considering the partition Π n by cubes, one can construct a b n -covering R n as in Theorem 5.13. This covering  (N, p). By regarding T as a family tree, it is straightforward that [T , o] has the same distribution as the part of [T , o ], up to the generation of the root (see [5] for more details on eternal family trees). Therefore, Lemma 5.7 of [5] implies that E 1/#(C n ∩K) = Finally, it remains to prove thatK is infinite a.s. when p = b −k . In this case, consider the eternal Galton-Watson tree [T , o ] as above. Proposition 6.8 of [5] implies that the generation of the root is infinite a.s. This proves the claim.

Cayley Graphs
As mentioned in Subsection 3.6, the dimension of a Cayley graph depends only on the group and not on the generating set. The following result connects it to the volume growth rate of the group. Note that Gromov's theorem [27] implies that the polynomial growth degree exists and is either an integer or infinity. Proof. First, assume α < ∞. The result of Bass [10] implies that there are constants c, C > 0 such that ∀r ≥ 1 : cr α < #N r (o) ≤ Cr α , where o is an arbitrary element of H. Second, assume α = ∞. The result of [50] shows that for any β < ∞, #N r (o) > r β for sufficiently large r. Therefore, part (i) of Lemma 5.4 implies that udim M (H) ≥ β. Hence, udim M (H) = udim H (H) = ∞ and the claim is proved.
It is natural to expect that M α (H) > 0 as well, but only a weaker inequality will be proved in Proposition 7.18.

Notes and Bibliographical Comments
The proof of Proposition 6.4 was suggested by R. Lyons. Bibliographical comments on some of the examples discussed in this section can be found at the end of Section 4. The example defined by digit restriction (Subsection 6.5.1) is inspired by an example in the continuum setting (see e.g., Examples 1.3.2 of [14]). The randomized discrete Cantor set (Subsection 6.5.2) is inspired by the random cantor set (see e.g., Section 3.7 of [14]).

Frostman's Theory
This section provides a unimodular version of Frostman's lemma and some of its applications. In a sense to be made precise later, this lemma gives converses to the mass distribution principle. It is a powerful tool in the theoretical analysis of the unimodular Hausdorff dimension. For example, it is used in this section to derive inequalities for the dimension of product spaces and embedded spaces (Subsections 7.4. 2 and 7.4.3). It is also the basis of many of the results in [6].

Unimodular Frostman Lemma
The statement of the unimodular Frostman lemma requires the definition of weighted Hausdorff content. The latter is based on the notion of equivariant weighted collections of balls as follows. For this, the following mark space is needed. Let Ξ be the set of functions c : R ≥0 → R ≥0 which are positive in only finitely many points; i.e., c −1 ((0, ∞)) is a finite set. Remark 7.9 below defines a metric on Ξ, so that the notion of Ξ-valued equivariant processes (Definition 2.9) is well defined. Such a process c is called an equivariant weighted collection of balls 2 . Consider a unimodular discrete space For v ∈ D, the reader can think of the value c r (v) := c(v)(r), if positive, to indicate that there is a ball in the collection, with radius r, centered at v, and with cost (or weight) c r (v). Note that extra randomness is allowed in the definition. A ball-covering R can be regarded a special case of this construction by letting c r (v) be 1 when r = R(v) and 0 otherwise.
It is straightforward that every equivariant ball-covering of Definition 3.14 gives a In the next theorem, to be consistent with the setting of the paper, the following notation    Let r ≥ M be fixed. For all discrete spaces D, let S := S D := {v ∈ D : w(N r (v)) > r α }. By the definition of S D , one has (i) for every edge e, one has 0 ≤ f (e) ≤ c(e), (ii) for every vertex v ∈ T \ L, one has Also, an equivariant cut-set is an equivariant subset Π of the edges of [T ; c] that separates the set of leaves L from the end in T .
Note that extra randomness is allowed in the above definition. The reader can think of the value f (v, F (v)) as the flow from v to F (v). So (7.7) can be interpreted as conservation of flow at the vertices except the leaves. Also, the leaves are regarded as the sources of the flow.
Since the number of leaves is infinite a.s., the sum of the flows exiting the leaves might be infinite. In fact, it can be seen that unimodularity implies that the sum is always infinite a.s. The idea is to use unimodularity to quantify how large is the flow. Similarly, in any equivariant cut-set, the sum of the conductances of the edges is infinite a.s. Unimodularity is also used to quantify the conductance of an equivariant cut-set. These are done in Definition 7.12 below.
Below, since each edge of T can be uniquely represented as (v, F (v)), the following convention is helpful. Convention 7.11. For the vertices v of T , the symbols f (v) and c(v) are used as abbreviations for f (v, F (v)) and c(v, F (v)), respectively. Also, by v ∈ Π, one means that the edge (v, F (v)) is in Π. An equivariant cut-set Π is called equivariantly minimal if there is no other equivariant cut-set which is a subset of Π a.s. If so, it can be seen that it is almost surely minimal as well; i.e., in almost every realization, it is a minimal cut set (see Lemma C.6 of [7]). In addition, if Π is minimal, then equality holds in the left inequality.
Proof. One can always consider an independent coupling of f and Π (as in the proof of Theorem 5.2). So assume (f , Π) is equivariant from the beginning. Note that the whole construction (with conductances, the flow and the cut-set) is unimodular (Lemma 2.12).
For every leaf v ∈ L, let τ (v) be the first ancestor of v such that (v, F (v)) ∈ Π. Then, send mass f (v) from each leaf v to τ (v). By the mass transport principle (2.2), one gets where the last equality holds because f is a flow. Moreover, if Π is minimal, then the above inequality becomes an equality and the claim follows.
The main result is the following converse to the above lemma. holds. The same proof works for this case as well. This will be used in Subsection 7.3.
Proof of Theorem 7.14. For n ≥ 1, let T n be the sub-forest of T obtained by keeping only vertices of height at most n in T . Each connected component of T n is a finite tree which contains some leaves of T . For each such component, namely T , do the following: if T has more than one vertex, consider the maximum flow on T between the leaves and the top vertex (i.e., the vertex with maximum height in T ). If there is more than one maximum flow, choose one of them randomly and uniformly. Also, choose a minimum cut-set in T randomly and uniformly. Similarly, if T has a single vertex v, do the same for the subgraph with vertex set {v, F (v)} and the single edge adjacent to v. By doing this for all components of T n , a (random) function f n on the edges and a cut-set Π n are obtained (by letting f n be zero on the other edges). Π n is always a cut-set, but f n is not a flow. However, f n satisfies (7.7) for vertices of T n \ L, except the top vertices of the connected components of T n . Also, it can be seen that f n and Π n are equivariant.
For each component T of T n , the set of leaves of T , excluding the top vertex, is L ∩ T . So the max-flow min-cut theorem of Ford-Fulkerson [25] (see e.g., Theorem 3.1.5 of [14]) gives that, for each component T of T n , one has Since 0 ≤ f n (·) ≤ c n (·), one can see that the distributions of f n are tight (see [7]). Therefore, there is a sequence n 1 , n 2 , . . . and an equivariant process f such that f ni → f (weakly). It is not hard to deduce that f is a legal equivariant flow. Also, since f (o) and 1 {o∈L} are continuous functions of [T , o; f ] and their product is bounded (by the assumption on c), one gets that Therefore, max f |f | ≥ inf Π c(Π). Note that the maximum of |f | is attained by the same tightness argument as above. So Lemma 7.13 implies that equality holds and the claim is proved.

A Unimodular Frostman Lemma for Point Processes
In the Euclidean case, another form of the unimodular Frostman lemma is given below. Its proof is based on the max-flow min-cut theorem of Subsection 7.2. As will be seen, the claim implies that in this case, Conjecture 7.4 holds up to a constant factor (Corollary 7.17). However, the weight function obtained in the theorem needs extra randomness.
Theorem 7.16. Let Φ be a point-stationary point process in R k endowed with the l ∞ metric, and let α ≥ 0. Then, there exists an equivariant weight function w on Φ such that, almost surely, ∀v ∈ Φ, ∀r ≥ 1 : w(N r (v)) ≤ r α In particular, if H α 1 (Φ) > 0, then w(0) is not identical to zero.
A similar result holds for the Euclidean metric or other equivalent metrics by just changing the constant 3 −k in (7.9). In the following proof, Φ is regarded as a counting measures; i.e., for all A ⊆ R d , Φ(A) := #(Φ ∩ A).
Proof. Let b > 1 be an arbitrary integer (e.g., b = 2). x For every integer n ≥ 0, let Q n be the stationary partition of R k by translations of the cube [0, b n ) k as in Subsection 5.3. Consider the nested coupling of these partitions for n ≥ 0 (i.e., every cube of Q n is contained in some cube of Q n+1 for every n ≥ 0) independent of Φ. Let T 0 be the tree whose vertices are the cubes in ∪ n Q n and the edges are between all pairs of nested cubes in Q n and Q n+1 for all n. Let T ⊆ T 0 be the subtree consisting of the cubes q n (v) for all v ∈ Φ and n ≥ 0. The set L of the leaves of T consists of the cubes q 0 (v) for all v ∈ Φ. Let σ := q 0 (0) ∈ L. Note that in the correspondence v → q 0 (v), each cube σ ∈ L corresponds to Φ(σ) ≥ 1 points of Φ. Therefore, by verifying the mass transport principle, it can be seen that the distribution of [L, σ], biased by 1/Φ(σ), is unimodular; i.e., for every measurable g ≥ 0. In addition, g can be allowed to depend on T in this equation (but the sum is still on σ ∈ L). Therefore, one can assume the metric on L is the graphdistance metric induced from T (see Theorem 3.31). Moreover, Theorem 5 of [35] implies that by a further biasing and choosing a new root for T , one can make T unimodular. More precisely, the following (possibly infinite) measure on D * is unimodular: where e n := Φ(q n (0)). Let E denote the integral operator w.r.t. the measure P. For any equivariant flow f on T , the norm of f w.r.t. the measure P (see Remark 7.15) satisfies where the second equality is by (7.10). Consider the conductance function c(τ ) := b nα for all cubes τ of edge length b n in T and all n. Therefore, Theorem 7.14 and Remark 7.15 imply that the maximum of E [f (σ)] over all equivariant legal flows f on [T , σ] is attained (note that [T , σ] is not unimodular, but the theorem can be used for P). Denote by f 0 the maximum flow. Let w be the weight function on Φ defined by w(v) = δf 0 (q 0 (v))/Φ(q 0 (v)), for all v ∈ Φ, where δ := (b + 1) −k . The claim is that w satisfies the requirements (7.8) and (7.9). Since f 0 is a legal flow, it follows that for every cube σ ∈ T , one has w(σ) = δf 0 (σ) ≤ δc(σ) = δb nα .
Each cube σ of edge length r ∈ [b n , b n+1 ) in R k can be covered with at most (b + 1) k cubes of edge length b n in T 0 . If n ≥ 0, the latter are either in T or do not intersect Φ. So the above inequality implies that w(σ) ≤ r α . So (7.8) is proved for w.
To prove (7.9), given any equivariant cut-set Π of T , a covering of Φ can be constructed as follows: For each cube σ ∈ Π of edge length say b n , let τ (σ) be one of the points in σ ∩ Φ chosen uniformly at random and put a ball of radius b n centered at τ (σ). Note that this ball contains σ. Do this independently for all cubes in T . If a point in Φ is chosen more than once, consider only the largest radius assigned to it. It can be seen that this gives an equivariant covering of Φ, namely R. One has On the other hand, by (7.10), one can see that . Since Π is an arbitrary equivariant cut-set, by the unimodular max-flow min-cut theorem established above (Theorem 7.14) and the maximality of the flow f 0 , one gets that H α The following corollary shows that in the setting of Theorem 7.16, the claim of Conjecture 7.4 holds up to a constant factor (compare this with Lemma 7.8).
Proof. The claim is directly implied by (7.2), Theorem 7.16 and Remark 7.3.

Applications
The following subsections give some basic applications of the unimodular Frostman lemma. This lemma is also the basis of many results of [6].  . It is left to the reader to show that w is an equivariant weight function. One has w(N r (v 1 , v 2 )) = w 1 (N r (v 1 ))w 2 (N r (v 2 )) ≤ r α1+α2 . Also, by the independence assumption, w(o 1 , o 2 ) = 0 with positive probability. Therefore, the mass distribution principle (Theorem 5.2) implies that udim H (D 1 × D 2 ) ≤ α 1 + α 2 .
This proves the upper bound.

Notes and Bibliographical Comments
The unimodular Frostman lemma (Theorem 7.2) is analogous to Frostman's lemma in the continuum setting (see e.g., Thm 8.17 of [42]). The proof of Theorem 7.2 is also inspired by that of [42], but there are substantial differences. For instance, the proof of Lemma 7.8 and also the use of the duality of L 1 and L ∞ in the proof of Theorem 7.2 are new. The Euclidean version of the unimodular Frostman lemma (Theorem 7.16) and its proof are inspired by the continuum analogue (see e.g., [14]).
As already explained, the unimodular max-flow min-cut theorem (Theorem 7.14) is inspired by the max-flow min-cut theorem for finite trees. Also, the results and examples of Subsection 7.4.2 on product spaces are inspired by analogous in the continuum setting; e.g., Theorem 3.2.1 of [14].

Connections with Other Notions of Dimension
Several notions of dimension are already defined in the literature for discrete spaces in special cases. A few of them are listed in this subsection together with their connections to unimodular dimensions.
For subsets of Z d , the notions of upper and lower mass dimension are defined in [8], which are just the volume growth rates defined in Section 5. The paper [38] extends the upper mass dimension to general subsets A ⊆ R d and calls it the macroscopic Minkowski dimension of A (one may define lower macroscopic Minkowski dimension similarly). This extension is obtained by pixelizing A to get a subset of Z d . The unimodular Billingsley lemma states that for unimodular (i.e., point-stationary) and ergodic subsets of Z d , the unimodular Hausdorff dimension is between the upper and lower mass dimension. A similar result holds in the non-integer case as well: This is a direct corollary of Billingsley's lemma applied to the pixelization by a randomly-shifted lattice. It can also be proved by using weights in Billingsley's lemma similarly to the proof of Proposition 5.15.
Another notion is that of discrete (Hausdorff) dimension [8], which uses the idea behind the definition of the classical Hausdorff dimension by considering coverings of Φ ⊆ Z d by large balls and considering the cost ( r r+|x| ) α for each ball in the covering, where r and x are the radius and the center of the ball and α is a constant (in fact, this is a modified version of the definition of [8] mentioned in [14]). In the future work [6], it is shown that the discrete dimension is an upper bound for the unimodular Hausdorff dimension, when both notions are defined (i.e., for point-stationary point processes).
The unimodular Hausdorff dimension can be connected to the classical Hausdorff dimension via scaling limits. Such limits are random continuum metric spaces and can be defined by weak convergence w.r.t. the Gromov-Hausdorff-Prokhorov metric [37]. It is shown in the preprint [6] that if the unimodular discrete space admits a scaling limit, then the ordinary Hausdorff dimension of the limit is an upper bound for the unimodular Hausdorff dimension.
The above inequalities are expected to be equalities in most examples. The preprint [6] provides more discussion on the matter. Note that these comparison results imply relations between the volume growth rate, scaling limits and discrete dimension, which are of independent interest and which are new to the best of the authors' knowledge.
A problem of potential interest is the connection of unimodular dimensions to other notions of dimension. This includes Gromov's notion of asymptotic dimension [28], the spectral dimension of a graph (defined in terms of the return probabilities of the simple random walk), the typical displacement exponent of a graph (see [16] for both notions), the isoperimetric dimension of a graph [15], the resistance growth exponent of a graph, the stochastic dimension of a partition of Z d [11], etc. In statistical physics, one also assigns dimension and various exponents to finite models. Famous examples are self-avoiding walks and the boundaries of large percolation clusters.

Gauge Functions and the Unimodular Dimension Function
There exist unimodular discrete spaces D in which the udim H (D)-dimensional Hausdorff size is either zero or infinity (e.g., Examples 6.1 and 6.2). For such spaces, it is convenient to generalize the unimodular Hausdorff size as follows. Consider an increas- In addition, given a family of gauge functions (ϕ α ) α≥0 that is increasing in α and such that ∀α > β : lim r→∞ ϕ α (r)/ϕ β (r) = ∞, one can redefine the unimodular Hausdorff dimension by sup{α : M ϕα (D) = 0} (see e.g., the next paragraph). One can redefine the unimodular Minkowski dimension similarly. The authors have verified that the results of the paper can be extended to this setting except that Theorem 3.34 and the results of Subsection 3.8 require the doubling condition sup r≥1 ϕ(2r)/ϕ(r) < ∞. The general result of Subsection 4.1.2 can also be extended under the doubling condition. Also, the upper bounds in the unimodular mass distribution principle, the unimodular Billingsley lemma and the unimodular Frostman lemma hold in this more general setting (some other results require the doubling condition). However, for the ease of reading, the results are presented in the original setting of this paper.
As an example of the above framework, one can define the exponential dimension by considering ϕ α (r) := e αr . It might be useful for studying unimodular spaces with super-polynomial volume growth, which are more interesting in group theory (see Subsection 6.6). Other gauge functions may also be useful for groups of intermediate growth. Note that exponential gauge functions do not satisfy the doubling condition, and hence, the reader should be careful about using the results of this work for such gauge functions.

Negative Dimensions
If a compact metric space X is the union of k disjoint copies of 1 r X, then the similarity dimension of X is log k/ log r (see e.g., [14]). This definition can also be used for some infinite discrete sets as well. For instance, Z d is a union of 2 d copies of 2Z d . So, it can be said that the similarity dimension of Z d is negative. The (deterministic) discrete Cantor set (see e.g., [14]) is also (− log 2/ log 3)-dimensional. There are several further arguments, listed below, suggesting that one should actually assign negative dimensions to unimodular discrete spaces.
First, this would be natural in terms of definition. The unimodular Minkowski dimensions should be redefined by udim M (D) = growth (λ r ) and udim M (D) = growth (λ r ).
Using growth instead of decay would then unify the definition of the ordinary Minkowski dimension of compact sets and the unimodular Minkowski dimension. The former is microscopic (i.e., when r tends to 0), whereas the latter is macroscopic (r → ∞). One may also replace the unimodular Hausdorff by the negative of the definitions given so far.
Secondly, this unification of the definitions would also take care of the puzzling that the assumption of [26] on the tail of the jumps is not necessary for having only an upper bound). For the latter, one might try to get ideas from [48] (it is necessary to use intervals with different lengths). Another guess is that the image of the symmetric nearest-neighbor simple random walk in Z d is 2-dimensional when d ≥ 3. More generally, if the jumps are in the domain of attraction of a symmetric α-stable process, then the image is α-dimensional. These might be proved similarly to the analogous results in [9]. For the graph of the simple random walk equipped with the Euclidean metric (Subsection 6.4), the guess is that if the increments are in the domain of attraction of an α-stable distribution, where 0 < α ≤ 2, then udim M (Ψ) = udim H (Ψ) = min{1, max(0, 2α − 1)/α} (see Theorem 3.13 of [38]). Also, the guess is that the zero set of the symmetric nearest-neighbor simple random walk in Z 2 is 1 4 -dimensional. (see [29] or Theorem 5.5 of [20]). The guess is that there is no regularly varying dimension function (see [19]), except in the finite-variance case (α = 2), where one may guess that the dimension function is r 2 log log r (see [21]).

Drainage networks.
One can ask about the dimension of other drainage network models. In particular, the simple model of Subsection 4.5 can be extended to a model in Z k for k > 2 and the connected component containing the origin is unimodular. Note that in the case α = 0, the conjecture is implied by Proposition 3.28. Also, in the general case, the conjecture is implied by (7.13) and Conjecture 7.4. Another problem is the validity of Proposition 7.24 under the weaker assumption of being non-equivariantly embeddable. As a partial answer, if growth (#N r (o)) exists, then (7.12) holds. This is proved as follows: udim H (D 0 ) ≤ ess inf growth (#N r (o 0 )) ≤ ess inf growth (#N r (o)) = ess inf growth (#N r (o)) = udim H (D), where the first inequality and the last equality are implied by the unimodular Billingsley lemma.

List of conjectures and problems mentioned in the previous sections
It is not known whether the lower bound (4.3) for the Hausdorff dimension of unimodular one-ended trees is always an equality or not. Problem 5.12 asks whether the equality udim H (D) = growth (w(N r (o))) always holds. This is implied by Problem 5.21, which states that the upper and lower growth rates of w(N r (o)) (used in Billingsley's lemma) do not depend on w. It would be interesting to find connections between unimodular dimensions and other notions of dimension, some of which are discussed in Subsection 8.1. Also, as mentioned in the introduction and Subsection 8.2, the setting of this paper might be useful in the study of examples pertaining to statistical physics or group theory.