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

Nikos Frantzikinakis (1) , Emmanuel Lesigne (2) , Mate Wierdl (3)

1 Department of Mathematics PSU
2 LMPT - Laboratoire de Mathématiques et Physique Théorique
3 Department of Mathematics and Statistics

For every $k in N$, 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.

Mots clés: récurrence multiple

Domaines: Systèmes dynamiques [math.DS]

Nikos Frantzikinakis, Emmanuel Lesigne, Mate Wierdl. Sets of k-recurrence but not (k+1)-recurrence. Annales de l'Institut Fourier, 2006, 56 (4), pp.839-849. ⟨hal-00016961⟩