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

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

DOI

https://doi.org/10.1007/s00041-020-09807-w

ISSN

1069-5869

Type

Journal article

Rights

Copyright © 2021 The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1007/s00041-020-09807-w.

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.