Download Intersective polynomials and polynomial Szemeredi theorem - Bergelson, Vitaly; Leibman, Alexander; Lesigne, Emmanuel | ePub Online

Genre/Form: article in peer-reviewed journal Material Type: Internet resource Document Type: Internet Resource, Archival Material All Authors / Contributors: Bergelson, Vitaly; Leibman, Alexander; Lesigne, Emmanuel OCLC Number: 783905655 Language Note: ENG Abstract: Let $P=\p_1,\ld,p_r\\subset\Q[n_1,\ld,n_m]$ be a family of polynomials such that $p_i(\Zm)\sle\Z$, $i=1,\ld,r$. We say that the family $P$ has \it PSZ property\/ if for any set $E\sle\Z$ with $d*(E)=\limsup_N-M\ras\infty\frac E\cap[M,N-1] N-M0$ there exist infinitely many $n\in\Zm$ 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 $k\in\N$ there exists $n\in\Zm$ 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\'edi 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 $p_1,\ld,p_r\in\Q[n]$ are jointly intersective integral polynomials, then for any finite partition of\/ $\Z$, $\Z=\bigcup_i=1kE_i$, there exist $i\in\1,\ld,k\$ and $a,n\in E_i$ such that $\a,a p_1(n),\ld,a p_r(n)\\sln E_i$.