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Thursday, August 31, 2017

**Abstract:** Recently, Bourgain and Chang established a nonlinear Roth theorem in finite fields: any set (in a finite field) with not-too-small density contains many nontrivial triplets $x$, $x+y$, $x+y^2$. The key step in Bourgain-Chang's proof is a $1/10$-decay estimate of some bilinear form. We slightly improve the estimate to a $1/8$-decay (and thus a better lower bound for the density is obtained). Our method is also valid for 3-term polynomial progressions $x$, $x+P(y)$, $x+Q(y)$. Besides discrete Fourier analysis, algebraic geometry (theorems of Deligne and Katz) is used. This is a joint work with Xiaochun Li and Will Sawin.