Intersections of thick compact sets in ℝd

Kenneth Falconer, Alexia Yavicoli

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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.
Original languageEnglish
Pages (from-to)2291-2315
Number of pages25
JournalMathematische Zeitschrift
Volume301
Issue number3
Early online date17 Feb 2022
DOIs
Publication statusPublished - Jul 2022

Keywords

  • Thickness
  • intersections of measures
  • Patterns
  • Dimension
  • Schmidt games
  • Gap lemma

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