Arithmetic patches, weak tangents, and dimension
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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.
Fraser , J M & Yu , H 2017 , ' Arithmetic patches, weak tangents, and dimension ' Bulletin of the London Mathematical Society , vol Early View . DOI: 10.1112/blms.12112
Bulletin of the London Mathematical Society
© 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
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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