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

Frantzikinakis, N.; Lesigne, E.; Wierdl, M.

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

Authors

Frantzikinakis, N.

Lesigne, E.

Wierdl, M.

Description

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.
8 pages