Ergodic recurrence and bounded gaps between primes

TL;DR

This paper establishes that in measure-preserving systems, there are infinitely many prime gaps with bounded size where multiple recurrence properties occur simultaneously, extending classical recurrence results to prime numbers.

Contribution

It proves the existence of infinitely many bounded prime gaps exhibiting multiple recurrence in measure-preserving systems, linking ergodic theory with prime number distribution.

Findings

Existence of infinitely many prime tuples with bounded gaps and recurrence properties

Quantitative bounds on prime gaps depending on system parameters

Extension of multiple recurrence results to primes

Abstract

Let $(X,B_X,\mu,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in B_X$ 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 $$ \mu(A\cap T^{-(p_i-1)}A)\geq \mu(A)^2-\epsilon $$ 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 $\epsilon$.