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Arithmetic patches, weak tangents, and dimension

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Fraser_2017_Arithmetic_patches_BLMS_AAM.pdf (254.5Kb)
Date
02/2018
Author
Fraser, Jonathan MacDonald
Yu, Han
Keywords
Arithmetic progression
Arithmetic patch
Weak tangent
Assouad dimension
Szemerédi’s Theorem
Erdös-Turán conjecture
Steinhaus property
QA Mathematics
T-NDAS
BDC
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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.
Citation
Fraser , J M & Yu , H 2018 , ' Arithmetic patches, weak tangents, and dimension ' , Bulletin of the London Mathematical Society , vol. 50 , no. 1 , pp. 85-95 . https://doi.org/10.1112/blms.12112
Publication
Bulletin of the London Mathematical Society
Status
Peer reviewed
DOI
https://doi.org/10.1112/blms.12112
ISSN
0024-6093
Type
Journal article
Rights
© 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
Description
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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  • University of St Andrews Research
URI
http://hdl.handle.net/10023/11978

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