Intersections of thick compact sets in $\mathbb{R}^d$
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Publication:6359607
DOI10.1007/S00209-022-02992-YzbMath1514.11017arXiv2102.01186MaRDI QIDQ6359607
Alexia Yavicoli, Kenneth J. Falconer
Publication date: 1 February 2021
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 $ au$, there is a number $N( au)$ such that the set contains a translate of all sufficiently small similar copies of every set in $mathbb{R}^d$ with at most $N( au)$ elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.
Contents, measures, outer measures, capacities (28A12) Fractals (28A80) Arithmetic progressions (11B25) Hausdorff and packing measures (28A78)
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