Arithmetic patches, weak tangents, and dimension

Jonathan MacDonald Fraser; Han Yu

doi:10.1112/blms.12112

Arithmetic patches, weak tangents, and dimension

Jonathan MacDonald Fraser, Han Yu

Pure Mathematics

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

24 Citations (Scopus)

4 Downloads (Pure)

Abstract

We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including the presence (or lack of) arithmetic progressions (or patches in dimensions at least 2), the structure of tangent sets, and the Assouad dimension.

We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dimension is equivalent to having weak tangents with non-empty interior and to ‘asymptotically’ containing arbitrarily large arithmetic patches.

We present some applications of our results concerning sets of integers, which include a weak solution to the Erdös–Turán conjecture on arithmetic progressions.

Original language	English
Pages (from-to)	85-95
Journal	Bulletin of the London Mathematical Society
Volume	50
Issue number	1
Early online date	1 Nov 2017
DOIs	https://doi.org/10.1112/blms.12112
Publication status	Published - Feb 2018

Keywords

Arithmetic progression
Arithmetic patch
Weak tangent
Assouad dimension
Szemerédi’s Theorem
Erdös-Turán conjecture
Steinhaus property

Access to Document

10.1112/blms.12112

Fraser-2017-Arithmetic-patches-BLMS-AAM
© 2017, London Mathematical Society. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version 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/10.1112/blms.12112
Accepted author manuscript, 255 KB

Fractal Geometry and Dimension: Fractal Geometry and dimension theory
Fraser, J. (PI)
The Leverhulme Trust
1/09/16 → 30/06/18
Project: Fellowship

Jonathan Fraser
- jmf32st-andrews.acuk
- Pure Mathematics - Professor
Person: Academic

Cite this

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title = "Arithmetic patches, weak tangents, and dimension",

abstract = "We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including the presence (or lack of) arithmetic progressions (or patches in dimensions at least 2), the structure of tangent sets, and the Assouad dimension.We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dimension is equivalent to having weak tangents with non-empty interior and to {\textquoteleft}asymptotically{\textquoteright} containing arbitrarily large arithmetic patches.We present some applications of our results concerning sets of integers, which include a weak solution to the Erd{\"o}s–Tur{\'a}n conjecture on arithmetic progressions.",

keywords = "Arithmetic progression, Arithmetic patch, Weak tangent, Assouad dimension, Szemer{\'e}di{\textquoteright}s Theorem, Erd{\"o}s-Tur{\'a}n conjecture, Steinhaus property",

author = "Fraser, {Jonathan MacDonald} and Han Yu",

note = "The first named author is supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and the second named author is supported by a PhD scholarship provided bythe School of Mathematics in the University of St Andrews",

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AB - We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including the presence (or lack of) arithmetic progressions (or patches in dimensions at least 2), the structure of tangent sets, and the Assouad dimension.We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dimension is equivalent to having weak tangents with non-empty interior and to ‘asymptotically’ containing arbitrarily large arithmetic patches.We present some applications of our results concerning sets of integers, which include a weak solution to the Erdös–Turán conjecture on arithmetic progressions.

KW - Arithmetic progression

KW - Arithmetic patch

KW - Weak tangent

KW - Assouad dimension

KW - Szemerédi’s Theorem

KW - Erdös-Turán conjecture

KW - Steinhaus property

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JF - Bulletin of the London Mathematical Society

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ER -

Arithmetic patches, weak tangents, and dimension

Abstract

Keywords

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Fractal Geometry and Dimension: Fractal Geometry and dimension theory

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Jonathan Fraser

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