Abstract
The Khintchine recurrence theorem asserts that in a measure preserving system, for every set A and ε > 0, we have μ(A ∩ T−nA) ≥ μ(A)2 − ε for infinitely many n ∈ N. We show that there are systems having underrecurrent sets A, in the sense that the inequality μ(A ∩ T−nA) < μ(A)2 holds for every n ∈ N. In particular, all ergodic systems of positive entropy have under-recurrent sets. On the other hand, answering a question of V. Bergelson, we show that not all mixing systems have under-recurrent sets. We also study variants of these problems where the previous strict inequality is reversed, and deduce that under-recurrence is a much more rare phenomenon than over-recurrence. Finally, we study related problems pertaining to multiple recurrence and derive some interesting combinatorial consequences.
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Boshernitzan, M., Frantzikinakis, N. & Wierdl, M. Under-recurrence in the Khintchine recurrence theorem. Isr. J. Math. 222, 815–840 (2017). https://doi.org/10.1007/s11856-017-1606-8
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DOI: https://doi.org/10.1007/s11856-017-1606-8