Roth's theorem in many variables

2011/06/08 by Schoen, Tomasz, Shkredov, Ilya D.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1106.1601

Abstract

We prove, in particular, that if a subset A of 1, 2,..., N has no nontrivial solution to the equation x₁+x₂+x₃+x₄+x₅=5y then the cardinality of A is at most N e^-c(log N)^1/7-eps, where eps>0 is an arbitrary number, and c>0 is an absolute constant. In view of the well-known Behrend construction this estimate is close to best possible.

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