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

Jonathan Fraser; Pablo Shmerkin; Alexia Yavicoli

doi:10.1007/s00041-020-09807-w

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

Jonathan Fraser, Pablo Shmerkin, Alexia Yavicoli

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Article number	4
Number of pages	14
Journal	Journal of Fourier Analysis and Applications
Volume	27
Issue number	4
DOIs	https://doi.org/10.1007/s00041-020-09807-w
Publication status	Published - 10 Feb 2021

Keywords

Arithmetic progressions
Hausdorff dimension
Fractals

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10.1007/s00041-020-09807-w

Fraser-2021-Improved-bounds-on-the-JFAA-AAM
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.
Accepted author manuscript, 183 KB

https://arxiv.org/abs/1910.10074

Cite this

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title = "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 Szemer{\'e}di 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.",

keywords = "Arithmetic progressions, Hausdorff dimension, Fractals",

author = "Jonathan Fraser and Pablo Shmerkin and Alexia Yavicoli",

note = "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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AU - Fraser, Jonathan

AU - Shmerkin, Pablo

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N2 - 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.

AB - 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.

KW - Arithmetic progressions

KW - Hausdorff dimension

KW - Fractals

UR - https://www.scopus.com/pages/publications/85101026515

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DO - 10.1007/s00041-020-09807-w

M3 - Article

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VL - 27

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

IS - 4

M1 - 4

ER -

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

Abstract

Keywords

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New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this

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

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this