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Intersective polynomials and the polynomial Szemerédi theorem. (English) Zbl 1156.11007
The authors give a new ergodic proof of the polynomial Szemerédi theorem [{\it V. Bergelson} and {\it A. Leibman}, J. Am. Math. Soc. 9, No. 3, 725--753 (1996; Zbl 0870.11015)] which is an extension of {\it 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 $$\{n\in\Bbb Z,\ \exists a \mid \{a,a+ P_1(n),\dots, a+ P_r(n)\}\subset E\}$$ 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 $n\in\Bbb Z$ such that all $P_i(n)$ are divisible by $k$. Please note that reference [9] appeared [{\it N. Frantzikinakis}, Trans. Am. Math. Soc. 360, No. 10, 5435--5475 (2008; Zbl 1158.37006)] and that reference [11] appeared [{\it N. Frantzikinakis} et al., Proc. Lond. Math. Soc. (3) 98, No. 2, 504--530 (2009; Zbl 1173.37006)] (see also \url{arxiv:0711.3159}).

11B25Arithmetic progressions
37A45Relations of ergodic theory with number theory and harmonic analysis
Full Text: DOI
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