Finding linear patterns of complexity one

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We study the following generalization of Roth's theorem for 3-term arithmetic progressions. For s≥2, define a nontrivial s-configuration to be a set of s(s+1)/2 integers consisting of s distinct integers x1,⋯,xs as well as the averages (xi+xj)/2 (1≤i<j≤s). Our main result states that if a set A∩[N] has density δ > (log N)-c(s) for some positive constant c(s)>0 depending on s, then A contains a nontrivial s-configuration. This improves on the previous bound of the form δ > (log log N)-c(s) due to Dousse [3]. We also deduce, as a corollary, an improvement of a problem involving sum-free subsets.

Original languageEnglish
Pages (from-to)2311-2327
Number of pages17
JournalInternational Mathematics Research Notices
Issue number9
StatePublished - 2015

Bibliographical note

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© 2014 © The Author(s) 2014. Published by Oxford University Press. All rights reserved.

ASJC Scopus subject areas

  • General Mathematics


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