Multiple recurrence and convergence for certain averages along shifted primes

TL;DR

This paper proves that subsets of natural numbers with positive density contain specific polynomial patterns along shifted primes, using ergodic theory and recurrence results, and establishes convergence of related averages.

Contribution

It introduces new recurrence and convergence results for polynomial patterns along shifted primes, linking combinatorics and ergodic theory.

Findings

Positive density sets contain polynomial patterns along shifted primes

Established convergence of averages along primes

Applied ergodic theory to combinatorial number theory problems

Abstract

We show any subset $A\subset\mathbb{N}$ with positive upper Banach density contains the pattern $\{m,m+[n\alpha],\dots,m+k[n\alpha]\}$, for some $m\in\mathbb{N}$ and $n=p-1$ for some prime $p$, where $\alpha\in\mathbb{R}\backslash\mathbb{Q}$. Making use for the Furstenberg Correspondence Principle, we do this by proving an associated recurrence result in ergodic theory along the shifted primes. We also prove the convergence result for the associated averages along primes and indicate other applications of these methods.