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


For every k ∈ 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. We also point out a combinatorial consequence related to Szemerédi's theorem.

Publication Title

Annales de l'Institut Fourier