= Solution
Assume first that $\mu(A\cap T^{-n}A)=0$ for every $n\geq1$. For $n>m$ the <measure-preserving transformation> property gives
$$
\mu(T^{-m}A\cap T^{-n}A)=\mu(A\cap T^{-(n-m)}A)=0.
$$
Thus the sets $T^{-n}A$, $n\geq0$, are pairwise disjoint and all have measure $\mu(A)>0$, impossible in a probability space. Hence
$$
\boxed{\mu(A\cap T^{-n}A)>0\text{ for some }n>0.}
$$
The <Poincare recurrence theorem> states that almost every $x\in A$ returns to $A$ infinitely often. Let
$$
E=\{x\in A:T^nx\notin A\text{ for every }n\geq1\}.
$$
The sets $T^{-n}E$ are pairwise disjoint: if $x$ belonged to the $m$th and $n$th inverse images with $m<n$, then $T^mx\in E$ would return to $E\subseteq A$ after $n-m$ steps. Invariance and finiteness therefore give $\mu(E)=0$. A point of $A$ with only finitely many returns belongs, after its last return, to some $T^{-k}E$. The countable union of these null sets is null, proving the theorem.
Now let $S\subseteq\mathbb Z$ have $d^*(S)>0$. Put $x=\mathbf1_S\in\{0,1\}^{\mathbb Z}$ and let $\sigma$ be the bilateral shift. Choose intervals $I_j$ with $|I_j|\to\infty$ and $|S\cap I_j|/|I_j|\to d^*(S)$. A weak-star limit of
$$
\mu_j=\frac1{|I_j|}\sum_{m\in I_j}\delta_{\sigma^m x}
$$
exists on the compact shift space. Boundary terms show that the limit $\mu$ is $\sigma$-invariant. For the cylinder $A=\{y:y_0=1\}$,
$$
\mu(A)=d^*(S)>0.
$$
The stated polynomial recurrence theorem supplies $n>0$ with $\mu(A\cap\sigma^{-|P(n)|}A)>0$. Hence that cylinder intersection meets the orbit closure of $x$; because it is open, some shift $\sigma^m x$ lies in it. Therefore $m,m+|P(n)|\in S$. Orient these two integers according to the sign of $P(n)$ to obtain
$$
\boxed{a,b\in S\quad\text{and}\quad b-a=P(n).}
$$
This is the <Furstenberg correspondence principle>.
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