Sets of k-recurrence but not (k+1)-recurrence

N. Frantzikinakis, E. Lesigne, M. Wierdl

Published 2005-03-17, updated 2005-04-05Version 2

For every $k\in \mathbb{N}$, we produce a set of integers which is $k$-recurrent but not $(k+1)$-recurrent. This extends a result of Furstenberg who produced a 1-recurrent set which is not 2-recurrent. We discuss a similar result for convergence of multiple ergodic averages. Finally, we also point out a combinatorial consequence related to Szemer\' edi's theorem.