Related papers: Hausdorff dimension of affine random covering sets…

Let $E=\limsup\limits_{n\to\infty}(g_n+\xi_n)$ be the random covering set on the torus $\mathbb{T}^d$, where $\{g_n\}$ is a sequence of ball-like sets and $\xi_n$ is a sequence of independent random variables uniformly distributed on…

We consider linear mappings on the $d$-dimensional torus, defined by $T(x) = Ax \pmod 1$, where $A$ is an invertible $d \times d$ integer matrix, with no eigenvalues on the unit circle. In the case $d = 2$ and $\det A = \pm 1$, we give a…

We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we…

Let ${\pmb M}$, ${\pmb N}$ and ${\pmb K}$ be $d$-dimensional Riemann manifolds. Assume that ${\bf A}:=(A_n)_{n\in{\Bbb N}}$ is a sequence of Lebesgue measurable subsets of ${\pmb M}$ satisfying a necessary density condition and ${\bf…

Let $\{X_n= e^{2\pi i \theta_n}\}$ be a sequence of Steinhaus random variables, where $\theta_n$ are independent and uniformly distributed on $[0,1]$. We compute the almost sure Hausdorff dimension of the images and graphs of the random…

The almost sure Hausdorff dimension of the limsup set of randomly distributed rectangles in a product of Ahlfors regular metric spaces is computed in terms of the singular value function of the rectangles.

We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate…

In this paper, we study the uniform random covering problem in general metric space $(X,d)$. Let $\omega=(\omega_n)_{n\in\mathbb N}$ be a sequence of independent identically distributed random variables on $(X,\mu)$, and…

The Hausdorff dimension of the set of points that are covered infinitely many times by a sequence of randomly distributed balls in the unit cube can be expressed in terms of the sizes of the balls. This note presents a new proof of the…

The almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group is computed in terms of directed singular value functions.

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 show that, almost surely, the Hausdorff dimension $s_0$ of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension $k>s_0$. The result holds for random covering sets with a generating…

In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category.…

A formula for the Hausdorff dimension of typical limsup sets generated by randomly distributed isotropic rectangles in Heisenberg groups is derived in terms of directed singular value functions.

We calculate the almost sure Hausdorff dimension for a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous…

We calculate the almost sure Hausdorff dimension of uniformly random self-similar fractals. These random fractals are generated from a finite family of similarities, where the linear parts of the mappings are independent uniformly…

We consider the Hausdorff dimension of random covering sets generated by balls and general measures in Euclidean spaces. We prove, for a certain parameter range, a conjecture by Ekstr\"om and Persson concerning the exact value of the…

Let $A$ be an invertible $d\times d$ matrix with integer elements. Then $A$ determines a self-map $T$ of the $d$-dimensional torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Given a real number $\tau>0$, and a sequence $\{z_n\}$ of points in…

In this article, we estimate the Hausdorff dimension of dynamical coverings with respect to mixing ergodic systems. More precisely, if the ergodic measure is exact-dimensionnal, we establish a formula provided that the system is…

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in \Bbb R^{md}$, let $\pi^{{\bf a}}:\; \Sigma=\{1,\ldots, m\}^{\Bbb N}\to \Bbb R^d$ denote the…