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dc.contributor.authorFraser, Jonathan MacDonald
dc.contributor.authorYu, Han
dc.identifier.citationFraser , J M & Yu , H 2018 , ' Arithmetic patches, weak tangents, and dimension ' , Bulletin of the London Mathematical Society , vol. 50 , no. 1 , pp. 85-95 .
dc.identifier.otherPURE: 251246899
dc.identifier.otherPURE UUID: 6bda7c73-0d07-4fec-8300-33309c9a5ab4
dc.identifier.otherScopus: 85032803788
dc.identifier.otherORCID: /0000-0002-8066-9120/work/58285473
dc.identifier.otherWOS: 000424101200009
dc.descriptionThe 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 Andrewsen
dc.description.abstractWe 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.
dc.relation.ispartofBulletin of the London Mathematical Societyen
dc.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
dc.subjectArithmetic progressionen
dc.subjectArithmetic patchen
dc.subjectWeak tangenten
dc.subjectAssouad dimensionen
dc.subjectSzemerédi’s Theoremen
dc.subjectErdös-Turán conjectureen
dc.subjectSteinhaus propertyen
dc.subjectQA Mathematicsen
dc.titleArithmetic patches, weak tangents, and dimensionen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews.Pure Mathematicsen
dc.description.statusPeer revieweden

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