Some dynamical properties related to polynomials

Abstract

Let $d\in\mathbb{Z}$ and $p_i$ be an integral polynomial with $p_i(0)=0,1\leq i\leq d$. It is shown that if $S$ is thickly syndetic in $\mathbb{Z}$, then $\{(m,n)\in\mathbb{Z}^2:m+p_i(n),m+p_2(n),\ldots,m+p_d(n)\in S\}$ is thickly syndetic in $\mathbb{Z}^2$. Meanwhile, we construct a transitive, strong mixing and non-minimal topological dynamical system $(X,T)$, such that the set $\{x\in X:\forall\ \text{open}\ U\ni x,\exists\ n\in\mathbb{Z} \ \text{s.t.}\ T^{n}x\in U,T^{2n}x\in U\}$ is not dense in $X$.