I have found a statement in the introduction of the paper 'Sets of Recurrence and Generalized Polynomials' by Bergelson & Haland, which is

**Result:** Given a polynomial $p \in \mathbb{R}[x]$ such that $p(\mathbb{Z})\subseteq \mathbb{Z}$, the set $\{p(n) | n ∈ \mathbb{Z}\}$ is a set of recurrence iff $p(n)\equiv 0\;(\text{mod } a)$ has a solution for $n$ for each $a\in \mathbb{N}$,

where a set $S\subseteq \mathbb{Z}$ is called a set of recurrence if for any **invertible** measure preserving system $(X, \mathscr{B}, \mu, T )$ and any $A\in \mathscr{B}$ with $\mu(A)>0$ there exists $n\in S,n\neq 0$ such that $\mu(A\cap T^{-n}A) > 0$.

$\quad$ But there is another definition(for example the definition of 'Poincare sequence' in the book 'Recurrence in Ergodic Theory and Combinatorial Number Theory' by Furstenberg), which doesn't assume invertibility of the measure preserving system, i,e;

A set $S\subseteq \mathbb{Z}$ is called a set of recurrence if for **any** measure preserving system $(X, \mathscr{B}, \mu, T )$ and any $A\in \mathscr{B}$ with $\mu(A)>0$ there exists $n\in S,n\neq 0$ such that $\mu(A\cap T^{-n}A) > 0$.

$\quad$ My **question** is, if we consider the second definition for the 'set of recurrence', does the result still holds true? I mean is this a known result? If so, please share a reference. Thanks in advance.