Related papers: Hausdorff dimension, its properties, and its surpr…
In this paper we study the geometry of the attractors of holomorphic maps with an irrationally indifferent fixed point. We prove that for an open set of such holomorphic systems, the local attractor at the fixed point has Hausdorff…
We fix a positive integer $M$, and we consider expansions in arbitrary real bases $q>1$ over the alphabet $\{0,1,...,M\}$. We denote by $U_q$ the set of real numbers having a unique expansion. Completing many former investigations, we give…
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…
I. J. Good (1941) showed that the set of irrational numbers in $(0,1)$ whose partial quotients $a_n$ tend to infinity is of Hausdorff dimension $1/2$. A number of related results impose restrictions of the type $a_n\in B$ or $a_n\geq f(n)$,…
Recently many papers on cone metric spaces have been appeared, and main topological properties of such spaces have been obtained. A cone metric space is Hausdorff, and first countable, so the topology of it coincides with a topology induced…
In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that…
As for the remarkable study on the estimate of the Hausdorff dimension of a self-similar set due to weak contractions (Kitada A. et al. Chaos, Solitons & Fractals 13 (2002) 363-366), we present a mathematically simplified form which will be…
We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of…
We study parametrized families of orthogonal projections for which the dimension of the parameter space is strictly less than that of the Grassmann manifold. We answer the natural question of how much the Hausdorff dimension may decrease by…
The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$, ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to…
What does it mean for a shape to change continuously? Over the space of convex regions, there is only one "reasonable" answer. However, over a broader class of regions, such as the class of star-shaped regions, there can be many different…
The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any $0<\beta<\alpha$, any compact metric space $X$ of Hausdorff dimension $\alpha$…
If the system S of contracting similitudes on $ R^2$ satisfies open convex set condition, then the set F of extreme points of the convex hull $\tilde{K}$ of it's invariant self-similar set K has Hausdorff dimension 0 . If, additionally, all…
Considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the measure and dimension of $A+\Gamma:=\left\{a+v:a\in A, v\in \Gamma \right\}$ when $A\subset \mathbb{R}^2$ and…
We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In…
The classical Painlev\'e theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical…
Let us consider a sphere $S^{n-1}$ of radius $r$ in $\mathbb{R}^n$, where we have fixed poles $N$ and $S$. Suppose that $K$ is a set in $\mathbb{R}^n$ containing a translated copy of each meridian (that is an $S^{n-2}$-sphere) of $S^{n-1}$.…
Expansions in non-integer bases have been investigated abundantly since their introduction by R\'enyi. It was discovered by Erd\H{o}s et al. that the sets of numbers with a unique expansion have a much more complex structure than in the…
Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure $\mu$ vanishes, then the set of points where the principal value of the Cauchy singular integral of $\mu$ exists has…
Let $I=[0,1)$, $-1<\lambda<1$ and $f\colon I\to I$ be a piecewise $\lambda$-affine map of the interval $I$, i.e., there exist a partition $0=a_0<a_1<\cdots< a_{k-1}<a_k=1$ of the interval $I$ into $k\geq2$ subintervals and $b_1,\ldots,…