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Bounds for sets lacking x,x+y,x+y^2

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Let P_1,...,P_m be polynomials with zero constant term. Bergelson and Leibman’s generalization of Szemerédi’s theorem to polynomial progressions states that any subset A of [N] that lacks nontrivial progressions of the form x,x+P_1(y),\dots,x+P_m(y) satisfies |A|=o(N). Proving quantitative bounds in the Bergelson—Leibman theorem is an interesting and difficult generalization of the problem of proving bounds in Szemerédi’s theorem, and bounds are known only in a very small number of special cases. In this talk, I’ll discuss a bound for subsets of [N] lacking the progression x,x+y,x+y^2, which is the first progression of length at least three involving polynomials of differing degree for which a bound is known. This is joint work with Sean Prendiville.

This talk is part of the Discrete Analysis Seminar series.

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