Intersective polynomials and the polynomial Szemerédi theorem
The authors give a new ergodic proof of the polynomial Szemerédi theorem [\textit{V. Bergelson} and \textit{A. Leibman}, J. Am. Math. Soc. 9, No. 3, 725--753 (1996; Zbl 0870.11015)] which is an extension of \textit{E. Szemerédi's} theorem on arithmetic progressions [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)]. More precisely, a family of polynomials \(P_1,\dots, P_r\) that send integers to integers is said to have the PSZ property if for any subset \(E\) of integers with positive upper Banach density, the set \[ \{n\in\mathbb Z,\;\exists a \mid \{a,a+ P_1(n),\dots, a+ P_r(n)\}\subset E\} \] is not empty. The main result in this paper is that a family of polynomials as above has the PSZ property if and only if for any nonnegative integer \(k\) there exists \(n\in\mathbb Z\) such that all \(P_i(n)\) are divisible by \(k\). Please note that reference [9] and that reference [11] appeared [\textit{N. Frantzikinakis}, Trans. Am. Math. Soc. 360, No. 10, 5435--5475 (2008; Zbl 1158.37006)] and [\textit{N. Frantzikinakis} et al., Proc. Lond. Math. Soc. (3) 98, No. 2, 504--530 (2009; Zbl 1173.37006)] (see also \url{arxiv:0711.3159}).
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