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 that send integers to integers is said to have the PSZ property if for any subset of integers with positive upper Banach density, the set
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 there exists such that all are divisible by .
Please note that reference  appeared [N. Frantzikinakis, Trans. Am. Math. Soc. 360, No. 10, 5435–5475 (2008; Zbl 1158.37006)] and that reference  is available at