Ergodic recurrence and bounded gaps between primes

Abstract

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