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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 ${P}_{1},\cdots ,{P}_{r}$ 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

$\left\{n\in ℤ,\phantom{\rule{4pt}{0ex}}\exists a\mid \left\{a,a+{P}_{1}\left(n\right),\cdots ,a+{P}_{r}\left(n\right)\right\}\subset E\right\}$

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 $n\in ℤ$ such that all ${P}_{i}\left(n\right)$ 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

arxiv:0711.3159.

##### MSC:
 11B25 Arithmetic progressions 37A45 Relations of ergodic theory with number theory and harmonic analysis