Multiple recurrence and convergence for certain averages along shifted primes

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.

References in corpus (1)

• The shifted primes and the multidimensional Szemeredi and polynomial van der Waerden theorems

Cited by in corpus (2)

• A spectral refinement of the Bergelson-Host-Kra decomposition and new multiple ergodic theorems
• Closest integer polynomial multiple recurrence along shifted primes