Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions

Jonathan Fraser, Pablo Shmerkin, Alexia Yavicoli

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Abstract

We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid ε-approximations of arithmetic progressions. Some of these estimates are in terms of Szemerédi bounds. In particular, we answer a question of Fraser, Saito and Yu (IMRN 14:4419–4430, 2019) and considerably improve their bounds. We also show that Hausdorff dimension is equivalent to box or Assouad dimension for this problem, and obtain a lower bound for Fourier dimension.
Original languageEnglish
Article number4
Number of pages14
JournalJournal of Fourier Analysis and Applications
Volume27
Issue number4
DOIs
Publication statusPublished - 10 Feb 2021

Keywords

  • Arithmetic progressions
  • Hausdorff dimension
  • Fractals

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