Powers of sequences and recurrence

Details Powers of sequences and recurrence Powers of sequences and recurrence

2007-11-20

arxiv.org/abs/0711.3159v2

Mathematics

Dynamical Systems

30 pages. Numerous small changes made. To appear in the Proceedings of the London Mathematical Society

Scientific paper

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.

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