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.
Annales de l'Institut Fourier
Frantzikinakis, N., Lesigne, E., & Wierdl, M. (2006). Sets of k-recurrence but not (k + 1)-recurrence. Annales de l'Institut Fourier, 56 (4), 839-849. https://doi.org/10.5802/aif.2202