arXiv subjects

Ondřej Zindulka

Publications and source records attributed to Ondřej Zindulka.

8 recordsLinked to original sources

arXiv2026

More on dominated and microscopic sets

Let $S$ be a family of sequences decreasing to zero. A set $E$ in a metric space is $S$-dominated if, for every $s\in S$, there exists a countable cover $\{E_n\}$ of $E$ such that $diam E_n<s_n$ for every $n$. We continue the study of families of dominated sets initiated in the previous paper. We look, e.g., into Cartesian products of dominated sets with strong measure and interaction of microscopic and porous sets. Based upon a tight relationship of Hausdorff measures and dominated sets, cardinal invariants of the ideals of dominated sets are determined and a related problem of Kwela is resolved.

math.CA↗

arXiv2026

Little lip of a typical function and $σ$-porosity of graphs

For a mapping $f\colon X\to Y$ between metric spaces the function $\text{lip} f\colon X\to[0,\infty]$ defined by $\text{lip} f(x)=\liminf_{r\to0}\frac{\text{diam} f(B(x,r))}{r}$ is termed the little lip function of $f$. We prove that, given a locally compact domain $Ω\subseteq\mathbb{R}^d$, for a typical continuous function $f\colon Ω\to\mathbb{R}$ the set $\{x\inΩ:\mbox{lip} f(x)>0\}$ has both Hausdorff and lower packing dimensions exactly $d-1$, while the set $\{x\inΩ:\text{lip} f(x)=\infty\}$ has non-$σ$ finite $(d{-}1)$-dimensional Hausdorff measure. This sharp result roofs previous results of Balogh and Csörnyei, Hanson and Buczolich, Hanson, Rmoutil and Zürcher. It follows, e.g., that a graph of a typical function $f\in C(Ω)$ is $σ$-strongly porous, and for a typical function $f\colon[0,1]\to[0,1]$ there are sets $A,B\subseteq[0,1]$ of lower packing and Hausdorff dimension zero, respectively, such that the graph of $f$ is contained in the set $A\times[0,1]\cup[0,1]\times B$.

math.CA↗

arXiv2026

Dominated sets, microscopic sets and Hausdorff measures

Let $S$ be a family of sequences of positive numbers that decrease to 0, let $X$ be a metric space and $A \subset X$. $A$ is said to be $S$-dominated if, for every $s\in S$, a countable cover $\{E_n\}$ of $E$ can be found such that $diam E_n < s_n$ for all $n$. We examine the family of all $S$-dominated sets, denoted by $\mathcal{D}(S)$. In particular, we examine the connections between $\mathcal{D}(S)$ and families of sets with zero Hausdorff measure for some gauges.

math.CA↗

arXiv2019

Strong measure zero in Polish groups

The notion of strong measure zero is studied in the context of Polish groups. In particular, the extent to which the theorem of Galvin, Mycielski and Solovay holds in the context of an arbitrary Polish group is studied. Hausdorff measure and dimension is used to characterize strong measure zero. The products of strong measure zero sets are examined. Sharp measure zero, a notion stronger that strong measure zero, is shown to be related to meageradditive sets in the Cantor set and Polish groups by a theorem very similar to the theorem of Galvin, Mycielski and Solovay.

math.GN↗

arXiv2018

Mapping Borel sets onto balls and self-similar sets by Lipschitz and nearly Lipschitz maps

If $X$ is an analytic metric space satisfying a very mild doubling condition, then for any finite Borel measure $μ$ on $X$ there is a set $N\subseteq X$ such that $μ(N)>0$, an ultrametric space $Z$ and a Lipschitz bijection $ϕ:N\to Z$ whose inverse is nearly Lipschitz, i.e., $β$-Hölder for all $β<1$. As an application it is shown that a Borel set in a Euclidean space maps onto $[0,1]^n$ by a nearly Lipschitz map if and only if it cannot be covered by countably many sets of Hausdorff dimension strictly below $n$. The argument extends to analytic metric spaces satisfying the mild condition. Further generalization replaces cubes with self-similar sets, nearly Lipschitz maps with nearly Hölder maps and integer dimension with arbitrary finite dimension.

math.CA↗

arXiv2018

Mapping Analytic sets onto cubes by little Lipschitz functions

A mapping $f:X\to Y$ between metric spaces is called \emph{little Lipschitz} if the quantity $$ \operatorname{lip}(f(x)=\liminf_{r\to0}\frac{\operatorname{diam} f(B(x,r))}{r} $$ is finite for every $x\in X$. We prove that if a compact (or, more generally, analytic) metric space has packing dimension greater than $n$, then $X$ can be mapped onto an $n$-dimensional cube by a little Lipschitz function. The result requires two facts that are interesing in their own right. First, an analytic metric space $X$ contains, for any $\varepsilon>0$, a compact subset $S$ that embeds into an ultrametric space by a Lipschitz map, and $\dim_P S\geq\dim_P X-\varepsilon$. Second, a little Lipschitz function on a closed subset admits a little Lipschitz extension.

math.CA↗

arXiv2012

Properties of functions with monotone graphs

A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y) < c d(x,z) for all x<y<z in X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It is shown, e.g., that such a function can be almost nowhere differentiable, but must be differentiable at a dense set, and that Hausdorff dimension of the graph of such a function is 1.

math.CA↗

arXiv2012

Hausdorff dimension of metric spaces and Lipschitz maps onto cubes

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than $k$ can be always mapped onto a $k$-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces. As an application we essentially answer a question of Urbański by showing that the transfinite Hausdorff dimension (introduced by him) of an analytic subset $A$ of a complete separable metric space is the integer part of $\dim_H A$ if $\dim_H A$ is finite but not an integer, $\dim_H A$ or $\dim_H A-1$ if $\dim_H A$ is an integer and at least $ω_0$ if $\dim_H A=\infty$.

math.CA↗