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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 [V. Bergelson and A. Leibman, J. Am. Math. Soc. 9, No. 3, 725–753 (1996; Zbl 0870.11015)] which is an extension of E. Szemerédi’s theorem on arithmetic progressions [Acta Arith. 27, 199–245 (1975; Zbl 0303.10056)]. More precisely, a family of polynomials P 1 ,,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,a{a,a+P 1 (n),,a+P r (n)}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 such that all P i (n) are divisible by k.

Please note that reference [9] appeared [N. Frantzikinakis, Trans. Am. Math. Soc. 360, No. 10, 5435–5475 (2008; Zbl 1158.37006)] and that reference [11] is available at


11B25Arithmetic progressions
37A45Relations of ergodic theory with number theory and harmonic analysis