Sarah Peluse

Bounds in the polynomial Szemer\'edi theorem

Let P_1,...,P_m be polynomials with integer coefficients and zero constant term. Bergelson and Leibman’s polynomial generalization of Szemer\'edi’s theorem states that any subset A of {1,...,N} that contains no nontrivial progressions x,x+P_1(y),...,x+P_m(y) must satisfy |A|=o(N). In contrast to Szemer\'edi's theorem, quantitative bounds for Bergelson and Leibman's theorem (i.e., explicit bounds for this o(N) term) are not known except in very few special cases. In this talk, I will discuss recent progress on this problem.