Related papers: Hausdorff dimension, projections, intersections, a…
We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid $\varepsilon$-approximations of arithmetic progressions. Some of these estimates are in terms of Szemer\'{e}di bounds. In…
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…
According to the Wik theorem, there exist massive Helson sets on the circle. In particular, they can be of Hausdorff dimension one. We extend Wik's result to the multidimensional case.
We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two subsets E and K of d-dimensional Euclidean space.
We prove that the extrinsic Hausdorff dimension is always greater than or equal to the intrinsic Hausdorff dimension in models of triangulated random surfaces with action which is quadratic in the separation of vertices. We furthermore…
In this manuscript the concept of hyperspace is revisited. The main purpose is to study hyperconvergence and continuity of orbital and limit set functions for semigroup action on completely regular space. Some general facts on Hausdorff and…
In this article, we study several aspects of the intersections of algorithmically random closed sets. First, we answer a question of Cenzer and Weber, showing that the operation of intersecting relatively random closed sets (with respect to…
The merit of projecting data onto linear subspaces is well known from, e.g., dimension reduction. One key aspect of subspace projections, the maximum preservation of variance (principal component analysis), has been thoroughly researched…
In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as…
In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable…
By appropriate scaling of coupling constants a one-parameter family of ensembles of two-dimensional geometries is obtained, which interpolates between the ensembles of (generalized) causal dynamical triangulations and ordinary dynamical…
Optimal quasiconformal dimension distortions bounds for subsets of the complex plane have been established by Astala. We show that these estimates can be improved when one considers subsets of the real line of arbitrary Hausdorff dimension.…
We construct a quasiconformal mapping of $n$-dimensional Euclidean space, $n \geq 2$, that simultaneously distorts the Hausdorff dimension of a nearly maximal collection of parallel lines by a given amount. This answers a question of…
We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t/2+\epsilon$, where $\epsilon>0$ depends only on $s$ and $t$. This improves the previously best known bound for $2s<t\le 1+\epsilon(s,t)$, in particular…
We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural `dimension pair'. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets…
We study the Macroscopic Hausdorff dimension of the upper and lower level sets of the Airy processes, following the general method developed in Khoshnevisan et al. \cite{KKX17}. For the Airy$_1$ process, the approach to macroscopic…
Let $X$ be a metric space and $BCl(X)$ the collection of nonempty bounded closed subsets of $X$. We show that Hausdorff distance $d_H$ belongs to a specific family of real-valued distances on $BCl(X)$, each of which can be expressed as the…
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 in…
Following in the footsteps of P. Erd\H{o}s and A. R\'enyi we compute the Hausdorff dimension of sets of numbers whose digits with respect to their $Q$-Cantor series expansions satisfy various statistical properties. In particular, we…
The set of badly approximable $m \times n $ matrices is known to have Hausdorff dimension $mn $. Each such matrix comes with its own approximation constant $c$, and one can ask for the dimension of the set of badly approximable matrices…