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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00016961
Contributor : Emmanuel Lesigne <>
Submitted on : Tuesday, April 3, 2007 - 10:44:04 AM
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### Citation

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

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