Intersective polynomials and the polynomial Szemerédi theorem

DOI10.1016/J.AIM.2008.05.008zbMATH Open1156.11007arXiv0710.4862OpenAlexW2950682123MaRDI QIDQ936621

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Publication date: 19 August 2008

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Abstract: Let $P=p_1,ld,p_{r}subsetQ[n_1,ld,n_m]$ be a family of polynomials such that $$, $i=1,ld,r$. We say that the family $P$ has {it PSZ property} if for any set $$ with $d^{*}(E)=limsu{p}_{N-Masinfty}frac|Ecap[M,N-1]|N-M>0$ there exist infinitely many $$ such that $E$ contains a polynomial progression of the form hbox{$a,a+p_1(n),ld,a+p_r(n)$}. We prove that a polynomial family $P=p_1,ld,p_r$ has PSZ property if and only if the polynomials $p_1,ld,p_r$ are {it jointly intersective}, meaning that for any $kinN$ there exists $$ such that the integers $p_1(n),ld,p_r(n)$ are all divisible by $k$. To obtain this result we give a new ergodic proof of the polynomial Szemer'{e}di 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 $p_1,ld,p_{r}inQ[n]$ are jointly intersective integral polynomials, then for any finite partition of $$, $$, there exist $iin1,ld,k$ and $a,nin{E}_i$ such that $a,a+p_1(n),ld,a+p_r(n)sln{E}_i$.

Full work available at URL: https://arxiv.org/abs/0710.4862

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