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On the Structure of Sets with Few Three-Term Arithmetic Progressions

By Ernie Croot


Fix a prime p � 3, and a real number 0 < α � 1. Let S ⊂ Fn p be any set with the least number of solutions to x + y = 2z (note that this means that x,z,y is an arithmetic progression), subject to the constraint that |S | � αpn. What can one say about the structure of such sets S? In this paper we show that they are “essentially” the union of a small number of cosets of some large-dimensional subspace of Fn p.

Year: 2010
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