Related papers: Projections of random covering sets

In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$ of numbers having a unique expansion in the (non-integer) base $q$, and showed that…

A first-order expansion of the $\mathbb{R}$-vector space structure on $\mathbb{R}$ does not define every compact subset of every $\mathbb{R}^n$ if and only if topological and Hausdorff dimension coincide on all closed definable sets.…

For integer $m\ge3$, we study the dynamical system $(\Lambda_m,\sigma_m)$ where $\Lambda_m$ is the set $\{w\in\{0,1\}^\mathbb{N}: w$ does not contain $0^m$ or $1^m\}$ and $\sigma_m$ is the shift map on $\{0,1\}^\mathbb{N}$ restricted to…

Denoting the Hausdorff dimension of the Fibonacci Hamiltonian with coupling $\lambda$ by $\mathrm{HD}_\lambda$, we prove that for all but countably many $\lambda$, the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian…

We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are…

This article is a study guide for ``On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane" by Orponen and Shmerkin. We begin by introducing Furstenberg set problem and exceptional set of projections and…

We fully resolve the Furstenberg set conjecture in $\mathbb{R}^2$, that a $(s, t)$-Furstenberg set has Hausdorff dimension $\ge \min(s+t, \frac{3s+t}{2}, s+1)$. As a result, we obtain an analogue of Elekes' bound for the discretized…

We study the fine scaling properties of sets satisfying various weak forms of invariance. For general attractors of possibly overlapping bi-Lipschitz iterated function systems, we establish that the Assouad dimension is given by the…

We investigate the Hausdorff dimension of level sets defined by digit growth rates in $\theta$-expansions, a generalization of regular continued fractions. For any $\alpha \geq 0$, we prove that the set \[ E_\theta(\alpha) = \left\{ x \in…

Let $\varphi_0$ be a smooth area-preserving diffeomorphism of a compact surface $M$ and let $\Lambda_0$ be a horseshoe of $\varphi_0$ with Hausdorff dimension strictly smaller than one. Given a smooth function $f:M\to \mathbb{R}$ and a…

Let $K$ be the attractor of a linear iterated function system (IFS) $S_j(x)=\rho_jx+b_j,$ $j=1,\cdots,m$, on the real line satisfying the generalized finite type condition (whose invariant open set $\mathcal{O}$ is an interval) with an…

We prove a large deviations principle for orthogonal projections of the unit ball $\mathbb{B}_p^n$ of $\ell_p^n$ onto a random $k$-dimensional linear subspace of $\mathbb{R}^n$ as $n\to\infty$ in the case $2<p\le \infty$ and for the…

The set B of geodesic rays avoiding a suitable obstacle in a complete negatively curved Riemannian manifold determines a spectrum S. While various properties of this spectrum are known, we define and study dimension functions on S in terms…

We study families $\Phi$ of coverings which are faithful for the Hausdorff dimension calculation on a given set $E$ (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset…

We determine the Hausdorff, packing and box-counting dimension of a family of self-affine sets generalizing Bara\'nski carpets. More specifically, we fix a Bara\'nski system and allow both vertical and horizontal random translations, while…

Under which conditions and with which distortions can we preserve the pairwise-distances of low-complexity vectors, e.g., for structured sets such as the set of sparse vectors or the one of low-rank matrices, when these are mapped in a…

It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(\pi_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$…

We describe the shrinking target problem for random iterated function systems which semi-conjugate to a random subshifts of finite type. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The…

In this work we are interested in the self--affine fractals studied by Gatzouras and Lalley and by the author which generalize the famous general Sierpinski carpets studied by Bedford and McMullen. We give a formula for the Hausdorff…

We extend the results previously published on exact packing dimensions of random recursive constructions to include constructions satisfying commonly occurring conditions. We remove the restrictive assumption that the diameter reduction…