Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions
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.
Citation
Fraser , J , Shmerkin , P & Yavicoli , A 2021 , ' Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions ' , Journal of Fourier Analysis and Applications , vol. 27 , no. 4 , 4 . https://doi.org/10.1007/s00041-020-09807-w
Publication
Journal of Fourier Analysis and Applications
Status
Peer reviewed
ISSN
1069-5869Type
Journal article
Description
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.Collections
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