The ACO Seminar (2017–2018)

Szemerédi's theorem on arithmetic progressions states that any subset of the integers of positive density contains arbitrarily long arithmetic progressions x,x+y,...,x+my with y nonzero. Bergelson and Leibman proved a generalization of Szemerédi's theorem for polynomial progressions x,x+P₁(y),...,x+P_m(y). In contrast to Szemerédi's theorem, quantitative bounds for Bergelson and Leibman's theorem are not known except in a few special cases. In particular, there are no general results for three-term progressions x,x+P(y),x+Q(y). In this talk, I will discuss the related problem of obtaining bounds for subsets of finite fields that lack nontrivial three-term polynomial progressions. It turns out that any subset A of F_q of size at least q^1-1/16+δ contains very close to |A|³/q progressions x,x+P(y),x+Q(y). I will show how to deduce this result from a bound for the dimension of a certain algebraic variety, and sketch how such a dimension bound can be proven.

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.