Under-recurrence in the Khintchine recurrence theorem
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.
Israel Journal of Mathematics
Boshernitzan, M., Frantzikinakis, N., & Wierdl, M. (2017). Under-recurrence in the Khintchine recurrence theorem. Israel Journal of Mathematics, 222 (2), 815-840. https://doi.org/10.1007/s11856-017-1606-8