Ergodic averages for sparse sequences along primes

TL;DR

This paper studies the behavior of multiple ergodic averages along prime numbers for sequences derived from smooth functions with polynomial growth, establishing new results in recurrence and additive combinatorics.

Contribution

It introduces a comparison between Cesàro averages and prime-weighted averages, proving new recurrence results along primes and confirming a conjecture of Frantzikinakis.

Findings

Positive density sets contain arithmetic progressions with prime-powered steps

Established ergodic averages convergence along primes for polynomial sequences

Proved a conjecture linking primes and polynomial progressions

Abstract

We investigate the limiting behavior of multiple ergodic averages along sparse sequences evaluated at prime numbers. Our sequences arise from smooth and well-behaved functions that have polynomial growth. Central to this topic is a comparison result between standard Ces\'{a}ro averages along positive integers and averages weighted by the (modified) von Mangoldt function. The main ingredients are a recent result of Matom\"{a}ki, Shao, Tao and Ter\"{a}v\"{a}inen on the Gowers uniformity of the latter function in short intervals, a lifting argument that allows one to pass from actions of integers to flows, a simultaneous (variable) polynomial approximation in appropriate short intervals, and some quantitative equidistribution results for the former polynomials. We derive numerous applications in multiple recurrence, additive combinatorics, and equidistribution in nilmanifolds along primes. In particular, we deduce that any set of positive density contains arithmetic progressions with step $\lfloor p^c \rfloor$, where $c$ is a positive non-integer and $p$ denotes a prime, establishing a conjecture of Frantzikinakis.