Sarah Peluse

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

Let P_1(y), ..., P_m(y) be polynomials with integer coefficients and zero constant term. Bergelson and Leibman's generalization of Szemerédi's theorem to polynomial progressions states that any subset A of [N] lacking nontrivial progressions of the form x, x+P_1(y), ..., 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.