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ARITHMETIC PROGRESSIONS IN SETS OF SMALL DOUBLING

Part of: Sequences and sets

Published online by Cambridge University Press:  17 February 2016

Kevin Henriot
Kevin Henriot*
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver BC V6T 1Z2, Canada email khenriot@math.ubc.ca

Abstract

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We show that if a finite, large enough subset $A$ of an arbitrary abelian group $G$ satisfies the small doubling condition $|A+A|\leqslant (\log |A|)^{1-{\it\varepsilon}}|A|$ , then $A$ must contain a three-term arithmetic progression whose terms are not all equal, and $A+A$ must contain an arithmetic progression or a coset of a subgroup, either of which is of size at least $\exp [c(\log |A|)^{{\it\delta}}]$ . This extends analogous results obtained by Sanders, and by Croot, Łaba and Sisask in the case where $G=\mathbb{Z}$ .

MSC classification

Primary: 11B13: Additive bases, including sumsets 11B30: Arithmetic combinatorics; higher degree uniformity

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 587 - 613
Copyright
Copyright © University College London 2016 

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