Powers of sequences and recurrence


We study recurrence and multiple recurrence properties along the kth 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 Mendès-France, which 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. © 2008 London Mathematical Society.

Publication Title

Proceedings of the London Mathematical Society