We generalize a theorem of Volkmann concerning the Hausdorff measures on subfields of $\R$. Our short proof is based on a mensural trichotomy law for invariant subsets of a locally compact group.

We prove the existence of three subspaces of $C[0,1]$, each homeomorphic to $C[0,1]$; the first consists only of infinitely many times differentiable functions, the second consists only of singular functions of bounded variation, and the third consists only of nowhere differentiable functions.

The purpose of this article is to show that if $-\infty < \alpha <\beta <\infty $, then the set $PD_{\alpha }^{\beta }$ of piecewise differentiable functions in $C([\alpha ,\beta ],{\mathbb R})$ is ${\bf \Pi }_{1}^{1}$-complete.

Basic limit theorems for the KH integral involve equiintegrable sets. We construct a family of Banach spaces $X_\Delta$ whose bounded sets are precisely the subsets of ${\cal KH}[0,1]$ that are equiintegrable and pointwise bounded. The resulting inductive limit topology on $\bigcup_\Delta X_\Delta = {\cal KH}[0,1]$ is barreled, bornological, and stronger than both pointwise convergence and the topology given by the Alexiewicz seminorm, but it lacks the countability and compatibility conditions that are often associated with inductive limits.

In this note we give, for any open subset in $\R^n$, a function describing the boundary of this set with exact regularity and being, globally, as regular as possible.

We consider a linear Smale-William's' horseshoe with different contraction/dilatation coefficients and find equilibrium states of maximal Hausdorff dimension. We compute this dimension and show an example when the state of maximal dimension is non-unique.

In this paper, we extend a theorem of Matejdes on quasicontinuous selections of upper Baire continuous set-valued mappings from compact (or separable) metric range spaces to regular $T_1$ range spaces. In addition, we also prove a quasicontinuous selection theorem for a special class of upper semicontinuous set-valued mappings.

Making use of ideas of Marczewski and Sierpinski we propose a general approach to studies on connections between measurability, continuity and relative continuity of functions. Theorem 2.1 shows that a well-known characterization of $(s)$-measurable Marczewski functions can be extended to the case of functions measurable with respect to a wide class of algebras involved with a topology. Theorem 2.2 gene\-ralizes the Denjoy-Stepanoff theorem and shows that the Denjoy-Stepanoff property stating the continuity of $\mc A$-measurable functions at all points of a co-negligible set is quite common while an algebra $\mc A$ and an ideal $\mc J$ are the results of operations $S$ and $S_0$ on $\tau\setminus\mc I$ for a given topology $\tau$. Also from the obtained results we conclude new theorems concerning the algebras associated with product ideals (Theorems \ref{t320} and \ref{t310}).

In this paper we continue the work started by Broomhead, Montaldi and Sidorov investigating the Hausdorff dimension of fat Sierpiński gaskets. We obtain generic results where the contraction rate $\lambda$ is in a certain region.

Every Borel subset $K$ of an interval $[c,d]$ induces a sequence of eigenvalues. If $K$ is closed, the asymptotic behavior of the eigenvalues is related to the positions and lengths of its complementary intervals. The rate of growth becomes ``lowest possible'' if $K$ has self-similarity properties. Eigenvalues of a vibrating string with singular mass distribution are eigenvalues associated with a set $K$.

This paper corrects some errors found in [2], which discusses an extension of Lorentz transformations over a non-Archimedean valued field; namely, the $p$-adic field $\mathbb{Q}_{p}$. The paper [2] is based on the results given by Hyers [7] which showed that for a continuous function $f$ defined on $\mathbb{R}$, the Cauchy functional equation $f(x + y) = f(x) + f(y)$ is stable. By stable we mean that if there exists $\epsilon >0$ such that $\| f(x + y) - f(x) - f(y) \| < \epsilon$, $\forall x,y$, then there exists a unique and continuous $\mathcal{L}$ such that $\| \mathcal{L}(x) - f(x) \| \leq \epsilon$, $\forall x$ and $\mathcal{L}(x + y) = \mathcal{L}(x) + \mathcal{L}(y)$. In this paper, we show this result is true on the $p$-adic field $\mathbb{Q}_p$.

Using almost-invariant sets, we show that a family of Marczewski--Burstin algebras over groups are not closed under algebraic sums. We also give an application of almost-invariant sets to the difference property in the sense of de~Bruijn. In particular, we show that if $G$ is a perfect Abelian Polish group then there exists a Marczewski null set $A \subseteq G$ such that $A+A$ is not Marczewski measurable, and we show that the family of Marczewski measurable real valued functions defined on $G$ does not have the difference property.

Recall that a Borel probability measure $\mu$ on $\R$ is called extremal if $\mu$-almost every number in $\R$ is not very well approximable. In this paper, we prove extremality (and implying it the exponentially fast decay property (efd)) of conformal measures induced by $1$-dimensional finite parabolic iterated function systems. We also investigate the doubling property of these measures and we estimate from below the Hausdorff dimension of the limit sets of such iterated systems.

A density continuous function is defined as a continuous function from a Tychonoff space $X$ into the real numbers with the density topology. The collection of density continuous functions on $X$ is denoted by $C(X,\mathbb{R}_d)$. It is shown that $C(X,\mathbb{R}_d)$ is a ring precisely when each density continuous function is locally constant, and in this case $X$ is defined to be a density $P$-space. Examples of density $P$-spaces are given.

In the present paper, some Olech and Visintin-type results are obtained in Henstock-Kurzweil-Pettis integration. More precisely, under extreme or denting point condition, one can pass from weak convergence (i.e. convergence with respect to the topology induced by the tensor product of the space of real functions of bounded variation and the topological dual of the initial Banach space) or from the convergence of integrals to strong convergence (i.e. in the topology of Alexiewicz norm or, even more, of Pettis norm). Our results extend the results already known in the Bochner and Pettis integrability setting.

We show that an analytic subset of the finite dimensional Euclidean space $\real^\dimens$ is purely unrectifiable if and only if the image of any of its compact subsets under every \locquo{} function is a Lebesgue null. We also construct purely unrectifiable compact sets of Hausdorff dimension greater than $1$ which are necessarily sent to finite sets by \locquo{} functions.

Given two topological spaces $X$ and $Y$ and a family ${\mathcal O}_\ast$ of subsets of $X$, a function $f: X \rightarrow Y$ is called ${\mathcal O}_\ast$-continuous if $f^{-1}(V) \in {\mathcal O}_\ast$ for every open set $V \subseteq Y$. An ${\mathcal O}_\ast$-step function is meant to be a function $\varphi: X \rightarrow Y$ that is piecewise constant on a partition of $X$ into sets from ${\mathcal O}_\ast$. Using some technical assumptions on $X$, $Y$, and ${\mathcal O}_\ast$ we give representations of ${\mathcal O}_\ast$-continuous functions as uniform limits of ${\mathcal O}_\ast$-step functions. We deal in particular with $\alpha$-continuous, nearly continuous, almost quasi-continuous, and somewhat continuous functions. The paper is motivated by a corresponding characterization of quasi-continuous functions.

A function $f : \mathbb{R}^n \to \mathbb{R}$ satisfies condition $(Q_{r,i}(x))$ (resp. ($Q_{r,s}(x))$, [$Q_{r,o}(x)$]) at a point $x$ if for each real $r > 0$ and for each set $U$ containing $x$ and belonging to Euclidean topology in $\mathbb{R} ^n$ (resp. to the strong density topology [to the ordinary density topology]) there is a regular domain $I$ such that int$(I) \cap U \neq \emptyset $, $f\!\restriction\!I$ is integrable in the sense of Riemann and $|\frac{1}{\mu (U\cap I)}\int_{U \cap I} f(t)\,dt - f(x)| < r$. These notions are particular cases of their analogues for the Lebesgue integral. In this article we compare these notions with the classical quasicontinuity and integral quasicontinuities

In the paper we present an exhaustive discussion of the relations between Darboux-like functions within the class of additive Sierpiński-Zygmund (SZ) functions. In particular, we give an example of an additive Sierpiński-Zygmund (SZ) injection $f : \mathbb{R} \to\mathbb{R}$ such that $f^{-1}$ is not an SZ function. Under the assumption that $\mathbb{R}$ cannot be covered by less than $\mathfrak{c}$-many meager sets we give examples of an additive SZ bijection $f : \mathbb{R} \to\mathbb{R}$ such that $f^{-1}$ is not SZ and of an additive injection $f : \mathbb{R} \to\mathbb{R}$ such that both $f$ and $f^{-1}$ are SZ.

It is known that for a first return system of paths $\{R_x:x\in[0,1]\}$, the right path systems $R^+$ ( the left path system $R^-$) is right ( is left ) continuous and $R$ satisfies the IIC property. In this paper we consider path systems that are continuous and satisfy IIC and investigate the possibility of containing first return path systems. We also study the effect of turbulence on trajectories by treating them as sequences.

Let $A_x$ stand for $x$-section of a set $A\subset2^\omega\times2^\omega$. We prove that any sequence $A_j\subset2^\omega\times2^\omega$, $j\in\omega$ of analytic sets, with uncountable $\limsup_{j\in H}A_x^j$ for all $x\in2^\omega$ and $H\in [\omega]^\omega$ admits a perfect set $P\subset2^\omega$ and $H\subset [\omega]^\omega$ with uncountable $\bigcap_{j\in H}A_x^j$ for all $x\in P$. This is a parametric version of the Komjath theorem [2].

A result of H. Steinhaus states that any Lebesgue measurable set $X\subset R$ with positive Lebesgue measure has a property that its difference set contains an open interval around zero. In this note we will prove a statement, which, in a sense, complements it.

Lévy's dragon curve [P. Lévy, Les courbes planes ou gauches et les surfaces composées de parties semblables au tout, J. Ecole Polytechn., 227-247, 249-291 (1938)] is a well-known self-similar planar curve with non-empty interior. We derive an arithmetic expression for the coordinate functions of Lévy's dragon curve, and show that the 3/2 -dimensional Hausdorff measure of the graph of each coordinate function is strictly positive and finite. This complements known dimensional results concerning the coordinate functions of space-filling curves of Peano and Hilbert. The proof is based on deriving suitable uniform upper bounds for the sizes of the graphs' level sets.

In this article I investigate the measurability of symmetrically quasicontinuous (in the sense of Piotrowski and Vallin) functions and symmetrically cliquish functions $f : R^2\to R$ with respect to the Euclidean topology and with respect to the density topology $T_d\times T_d$.