Joint ergodicity of fractional powers of primes

TL;DR

This paper proves mean convergence and multiple recurrence for averages involving fractional powers of primes, revealing new combinatorial patterns in dense sets of integers and connecting ergodic theory with number theory.

Contribution

It introduces a novel approach to joint ergodicity for sequences involving fractional powers of primes, combining ergodic theory with elementary sieve and equidistribution results.

Findings

Established mean convergence for fractional prime power averages.

Proved that dense sets contain specific fractional prime patterns.

Connected ergodic averages with prime number theory techniques.

Abstract

We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive upper density contains patterns of the form $\{m,m+[p_n^a], m+[p_n^b]\}$, where $a,b$ are positive non-integers and $p_n$ denotes the $n$-th prime, a property that fails if $a$ or $b$ is a natural number. Our approach is based on a recent criterion for joint ergodicity of collections of sequences and the bulk of the proof is devoted to obtaining good seminorm estimates for the related multiple ergodic averages. The input needed from number theory are upper bounds for the number of prime $k$-tuples that follow from elementary sieve theory estimates and equidistribution results of fractional powers of primes in the circle.