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

DOI10.1007/S00041-020-09807-WzbMATH Open1498.11037arXiv1910.10074OpenAlexW3127639217MaRDI QIDQ2657387

Pablo Shmerkin, Alexia Yavicoli, Jonathan M. Fraser

Publication date: 12 March 2021

Published in: The Journal of Fourier Analysis and Applications (Search for Journal in Brave)

Abstract: We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid $varepsilon$-approximations of arithmetic progressions. Some of these estimates are in terms of Szemer'{e}di bounds. In particular, we answer a question of Fraser, Saito and Yu (IMRN, 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.

Full work available at URL: https://arxiv.org/abs/1910.10074

zbMATH Keywords

Hausdorff dimensionfractalsarithmetic progressions

Mathematics Subject Classification ID

Fractals (28A80) Arithmetic progressions (11B25)

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Related Items (1)

Exposé Bourbaki 1196 : Recent progress on bounds for sets with no three terms in arithmetic progression (after Bloom and Sisask, Croot, Lev, and Pach, and Ellenberg and Gijswijt)

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