Intersections of thick compact sets in $\mathbb{R}^d$

Kenneth Falconer; Alexia Yavicoli

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arXiv · 2102.01186

Intersections of thick compact sets in $\mathbb{R}^d$

Kenneth Falconer·Alexia Yavicoli

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Abstract

We introduce a definition of thickness in $\mathbb{R}^d$ and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt's game. As an application we prove that given any compact set in $\mathbb{R}^d$ with thickness $\tau$, there is a number $N(\tau)$ such that the set contains a translate of all sufficiently small similar copies of every set in $\mathbb{R}^d$ with at most $N(\tau)$ elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.

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BibTeX RIS

Kenneth Falconer, Alexia Yavicoli. 2021-02-01. Intersections of thick compact sets in $\mathbb{R}^d$. https://arxiv.org/abs/2102.01186

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