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 . We also deduce, as a corollary, an improvement of a problem involving sum-free subsets.
|Number of pages||17|
|Journal||International Mathematics Research Notices|
|State||Published - 2015|
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© 2014 © The Author(s) 2014. Published by Oxford University Press. All rights reserved.
ASJC Scopus subject areas
- Mathematics (all)