Intersective polynomials and polynomial Szemeredi theorem

Let $P=\{p_1,\text{ld},p_r\}\subset \text{Q}[n_1,\text{ld},n_m]$ be a family of polynomials such that $p_i({\mathbb{Z}}^m)\text{sle}\mathbb{Z}$, $i=1,\text{ld},r$. We say that the family $P$ has {\it PSZ property} if for any set $E\text{sle}\mathbb{Z}$ with $d^{\ast }(E)={\mathrm{lim\hspace{0.17em}sup}}_{N-M\text{ras}\mathrm{\infty }}\frac{\mathrm{\mid }E\cap [M,N-1]\mathrm{\mid }}{N-M}>0$ there exist infinitely many $n\in {\mathbb{Z}}^m$ such that $E$ contains a polynomial progression of the form \hbox{$\{a,a+p_1(n),\text{ld},a+p_r(n)\}$}. We prove that a polynomial family $P=\{p_1,\text{ld},p_r\}$ has PSZ property if and only if the polynomials $p_1,\text{ld},p_r$ are {\it jointly intersective}, meaning that for any $k\in \mathbb{N}$ there exists $n\in {\mathbb{Z}}^m$ such that the integers $p_1(n),\text{ld},p_r(n)$ 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 $p_1,\text{ld},p_r\in \text{Q}[n]$ are jointly intersective integral polynomials, then for any finite partition of $\mathbb{Z}$, $\mathbb{Z}={\bigcup }_{i=1}^k{E}_i$, there exist $i\in \{1,\text{ld},k\}$ and $a,n\in E_i$ such that $\{a,a+p_1(n),\text{ld},a+p_r(n)\}\text{sln}{E}_i$.