Powers of sequences and recurrence

TL;DR

This paper investigates recurrence properties along powers of integers, showing independence between different powers and exploring connections with uniform distribution, motivated by number theory and extending classical recurrence results.

Contribution

It demonstrates that recurrence for certain powers does not constrain recurrence for others and proposes a conjecture linking higher-order recurrence with uniform distribution.

Findings

Recurrence along different powers is independent.

Recurrence properties relate to uniform distribution conjecture.

Motivates further research in higher-order recurrence and distribution.

Abstract

We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and Mend\`es-France, that links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher order recurrence and make a related conjecture.