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

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74539For 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 pagesDynamical SystemsCombinatorics37A45, 28D05Sets of k-recurrence but not (k+1)-recurrencetext