Closest integer polynomial multiple recurrence along shifted primes

TL;DR

This paper proves that certain polynomial recurrence properties along shifted primes occur in multidimensional settings, using ergodic theory and a transference principle to connect flows and integer actions.

Contribution

It extends polynomial multiple recurrence results to shifted primes and introduces a transference method for convergence in ergodic theory.

Findings

Parameters in multidimensional Szemerédi theorem intersect with shifted primes

Integer part polynomial convergence results established

Applications to Gowers uniform sets provided

Abstract

Following an approach presented by N. Frantzikinakis, B. Host and B. Kra, we show that the parameters in the multidimensional Szemer\'edi theorem for closest integer polynomials have non-empty intersection with the set of shifted primes $\mathbb{P}-1$ (or similarly of $\mathbb{P}+1$). Using the Furstenberg Correspondence Principle, we show this result by recasting it as a polynomial multiple recurrence result in measure ergodic theory. Furthermore, we obtain integer part polynomial convergence results by the same method, which is a transference principle that enables one to deduce results for $\mathbb{Z}$-actions from results for flows. We also give some applications of our approach on Gowers uniform sets.