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Proceedings of the London Mathematical Society Advance Access originally published online on October 7, 2008
Proceedings of the London Mathematical Society 2009 98(2):504-530; doi:10.1112/plms/pdn040
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© 2008 London Mathematical Society

Powers of sequences and recurrence

Nikos Frantzikinakis

Department of Mathematics
University of Memphis
Memphis, TN 38152
USA

Emmanuel Lesigne

Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 6083)
Université François Rabelais Tours
Fédération de Recherche Denis Poisson
Parc de Grandmont
37200 Tours
France
emmanuel.lesigne@lmpt.univ-tours.fr

Máté Wierdl

Department of Mathematics
University of Memphis
Memphis, TN 38152
USA
wierdlmate@gmail.com

Received 20 November 2007. Revision received 29 May 2008.

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.

2000 Mathematics Subject Classification 37A45 primary, 28D05, 05D10, 11B25 secondary.

The first author was partially supported by NSF grant DMS-0701027.


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