We show that some spectral properties of the almost Mathieu operator with frequency well approximated by rationals can be as poor as at all possible in the class of all one-dimensional discrete Schrödinger operators. For the case of critical coupling, we show that the Hausdorff measure of the spectrum may vanish (for appropriately chosen frequencies) whenever the gauge function tends to zero faster than logarithmically. For arbitrary coupling, we show that modulus of continuity of the integrated density of states can be arbitrary close to logarithmic; we also prove a similar result for the Lyapunov exponent as a function of the spectral parameter. Finally, we show that (for any coupling) there exist frequencies for which the spectrum is not homogeneous in the sense of Carleson, and, moreover, fails the Parreau–Widom condition. The frequencies for which these properties hold are explicitly described in terms of the growth of the denominators of the convergents.

For a 4-manifold M and a knot $k:{\mathbb{S}}^1\hookrightarrow \partial M$ with dual sphere $G:{\mathbb{S}}^2\hookrightarrow \partial M$, we compute the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings ${\mathbb{D}}^2\hookrightarrow M$ with boundary k using an invariant going back to Dax. Moreover, we construct a group structure on $\mathbb{D}(M;k)$ and show that it is usually neither abelian nor finitely generated. We recover all previous results for isotopy classes of spheres with framed duals and relate the group $\mathbb{D}(M;k)$ to the mapping class group of M.

We show that the v-sheaf local models of Scholze and Weinstein are unibranch. In particular, this proves that the scheme-theoretic local models defined in Anschütz, Gleason, Lourenço, and Richarz are always normal with reduced special fiber, thereby establishing the remaining cases of the geometric part of the Scholze–Weinstein conjecture when $p\le 3$. Our methods are general, topological, and simplify those of Zhu for tamely ramified groups in positive characteristic. As a technical input, we generalize a comparison theorem of nearby cycles of Huber to the v-sheaf setup.

We consider Bernoulli percolation on transitive graphs of polynomial growth. In the subcritical regime ($p<p_c$), it is well known that the connection probabilities decay exponentially fast. In the present paper, we study the supercritical phase ($p>p_c$) and prove the exponential decay of the truncated connection probabilities (probabilities that two points are connected by an open path, but not to infinity). This sharpness result was established by Chayes, Chayes, and Newman on ${\mathbb{Z}}^d$ and uses the difficult slab result of Grimmett and Marstrand. However, the techniques used there are very specific to the hypercubic lattices and do not extend to more general geometries. In this paper, we develop new robust techniques based on the recent progress in the theory of sharp thresholds and the sprinkling method of Benjamini and Tassion. On ${\mathbb{Z}}^d$, these methods can be used to produce a new proof of the slab result of Grimmett and Marstrand.