Intersections of thick compact sets in ℝd
Abstract
We introduce a definition of thickness in ℝd 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 ℝd with thickness τ, there is a number N(τ) such that the set contains a translate of all sufficiently small similar copies of every set in ℝd with at most N(τ) elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.
Citation
Falconer , K & Yavicoli , A 2022 , ' Intersections of thick compact sets in ℝ d ' , Mathematische Zeitschrift , vol. 301 , no. 3 , pp. 2291-2315 . https://doi.org/10.1007/s00209-022-02992-y
Publication
Mathematische Zeitschrift
Status
Peer reviewed
ISSN
0025-5874Type
Journal article
Rights
Copyright © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Description
Funding: Alexia Yavicoli was financially supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.Collections
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