Ergodic recurrence and bounded gaps between primes

Let $(X,B_X,μ,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in B_X$ with $μ(A)>0$ and $ε>0$. 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 $μ(A\cap T^{-(p_i-1)}A)\geq μ(A)^2-ε$ for each $0\leq i\leq m$ and $p_m-p_0<C_m,$ where $C_m>0$ is a constant only depending on $m$, $A$ and $ε$.

No reviews yet.

No comments yet.