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arXiv · 2603.20050

Dominated sets, microscopic sets and Hausdorff measures

Ondřej Zindulka·Piotr Nowakowski
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Abstract

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.

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Ondřej Zindulka, Piotr Nowakowski. 2026-03-20. Dominated sets, microscopic sets and Hausdorff measures. https://arxiv.org/abs/2603.20050

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