Uniform Distribution of Prime Powers and sets of Recurrence and van der Corput sets in Z^k

TL;DR

This paper introduces new results on recurrence and van der Corput sets in multi-dimensional integer spaces, using prime power equidistribution to unify and refine previous findings in the field.

Contribution

It presents a general equidistribution theorem involving prime powers that advances understanding of recurrence and van der Corput sets in Z^k.

Findings

Refined criteria for recurrence sets in Z^k.

Unified framework for previous results by Sarkozy, Furstenberg, and others.

New equidistribution results involving prime powers.

Abstract

We establish new results on sets of recurrence and van der Corput sets in Z^k which refine and unify some of the previous results obtained by Sarkozy, Furstenberg, Kamae and Mendes France, and Bergelson and Lesigne. The proofs utilize a general equidistribution result involving prime powers which is of independent interest.