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 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.
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Communicated by Alex Iosevich.
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JMF is financially supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034). PS is supported by a Royal Society International Exchange Grant and by Project PICT 2015-3675 (ANPCyT). AY is financially supported by the Swiss National Science Foundation, Grant No. P2SKP2_184047.
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Fraser, J.M., Shmerkin, P. & Yavicoli, A. Improved Bounds on the Dimensions of Sets that Avoid Approximate Arithmetic Progressions. J Fourier Anal Appl 27, 4 (2021). https://doi.org/10.1007/s00041-020-09807-w
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DOI: https://doi.org/10.1007/s00041-020-09807-w