Ergodic recurrence and bounded gaps between primes

Abstract:

Let $(\mathcal {X},\mathscr {B}_\mathcal {X},\mu ,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in \mathscr {B}_\mathcal {X}$ with $\mu (A)>0$ and $\epsilon >0$. For any $m\geq 1$, there exist infinitely many primes $p_0,p_1,\ldots ,p_m$ with $p_0<\cdots <p_m$ such that \begin{equation*} \mu (A\cap T^{-(p_i-1)}A)\geq \mu (A)^2-\epsilon \end{equation*} for each $0\leq i\leq m$ and \begin{equation*} p_m-p_0<C_{m,A,\epsilon }, \end{equation*} where $C_{m,A,\epsilon }>0$ is a constant only depending on $m$, $A$ and $\epsilon$.