Intersections of thick compact sets in ℝd

Kenneth Falconer; Alexia Yavicoli

doi:10.1007/s00209-022-02992-y

Intersections of thick compact sets in ℝ^d

Kenneth Falconer, Alexia Yavicoli

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We introduce a definition of thickness in 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 ℝ^d 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 ℝ^d 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.

Original language	English
Pages (from-to)	2291-2315
Number of pages	25
Journal	Mathematische Zeitschrift
Volume	301
Issue number	3
Early online date	17 Feb 2022
DOIs	https://doi.org/10.1007/s00209-022-02992-y
Publication status	Published - Jul 2022

Keywords

Thickness
intersections of measures
Patterns
Dimension
Schmidt games
Gap lemma

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10.1007/s00209-022-02992-yLicence: CC BY

2022-M.Zeit-Yavicoli-Intersections
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/.
Final published version, 575 KBLicence: CC BY

https://arxiv.org/abs/2102.01186

Cite this

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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. ",

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author = "Kenneth Falconer and Alexia Yavicoli",

note = "Funding: Alexia Yavicoli was financially supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.",

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AU - Yavicoli, Alexia

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PY - 2022/7

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N2 - 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.

AB - 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.

KW - Thickness

KW - intersections of measures

KW - Patterns

KW - Dimension

KW - Schmidt games

KW - Gap lemma

UR - https://arxiv.org/abs/2102.01186

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DO - 10.1007/s00209-022-02992-y

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VL - 301

SP - 2291

EP - 2315

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3

ER -

Intersections of thick compact sets in ℝ^d

Abstract

Keywords

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