Elsevier

Advances in Mathematics

Volume 219, Issue 1, 10 September 2008, Pages 369-388
Advances in Mathematics

Intersective polynomials and the polynomial Szemerédi theorem

https://doi.org/10.1016/j.aim.2008.05.008Get rights and content
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Abstract

Let P={p1,,pr}Q[n1,,nm] be a family of polynomials such that pi(Zm)Z, i=1,,r. We say that the family P has the PSZ property if for any set EZ with d(E)=lim supNM|E[M,N1]|NM>0 there exist infinitely many nZm such that E contains a polynomial progression of the form {a,a+p1(n),,a+pr(n)}. We prove that a polynomial family P={p1,,pr} has the PSZ property if and only if the polynomials p1,,pr are jointly intersective, meaning that for any kN there exists nZm such that the integers p1(n),,pr(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If p1,,prQ[n] are jointly intersective integral polynomials, then for any finite partition Z=i=1kEi of Z, there exist i{1,,k} and a,nEi such that {a,a+p1(n),,a+pr(n)}Ei.

Keywords

Polynomial Szemerédi theorem
Intersective polynomials
Nilmanifolds

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The first two authors were supported by NSF grant DMS-0600042.