Bounds in the polynomial Szemeredi theorem via Fourier analysis

Let $P_1,\ldots, P_m$ be polynomials with integer coefficients and zero constant term. Bergelson and Leibman’s polynomial generalization of Szemerédi’s theorem states that any subset $A$ of $\{1,\ldots,N\}$ that contains no nontrivial progressions $x, x+P_1(y), \ldots, x+P_m(y)$ must satisfy $|A|=o(N)$. In contrast to Szemerédi’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, focusing on arguments involving Fourier analysis.

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