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

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Date
10/02/2021
Author
Fraser, Jonathan
Shmerkin, Pablo
Yavicoli, Alexia
Funder
EPSRC
The Leverhulme Trust
Grant ID
EP/R015104/1
RPG-2019-034
Keywords
Arithmetic progressions
Hausdorff dimension
Fractals
QA Mathematics
T-NDAS
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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.
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.
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  • University of St Andrews Research
URL
https://arxiv.org/abs/1910.10074
URI
http://hdl.handle.net/10023/24913

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