Topological dynamical systems induced by polynomials and combinatorial consequences

Let $d\in \mathbb{N}$ and $p_i$ be an integral polynomial with $p_i(0)=0$, $1\le i\le d$. It is shown that if $S$ is piecewise syndetic in $\mathbb{Z}$, then $$\{(m,n)\in {\mathbb{Z}}^2:m+p_1(n),\dots ,m+p_d(n)\in S\}$$ is piecewise syndetic in ${\mathbb{Z}}^2$, which extends the result by Glasner and Furstenberg for linear polynomials. Our result is obtained by showing the density of minimal points of a dynamical system of ${\mathbb{Z}}^2$ action associated with the piecewise syndetic set $S$ and the polynomials $\{p_1,\dots ,p_d\}$. Moreover, it is proved that if $(X,T)$ is minimal, then for each non-empty open subset $U$ of $X$, there is $x\in U$ with $\{n\in \mathbb{Z}:T^{p_1(n)}x\in U,\dots ,T^{p_d(n)}x\in U\}$ piecewise syndetic.