Roth's theorem in many variables
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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