2016/03/24 by Boshernitzan, Michael, Frantzikinakis, Nikos, Wierdl, Máté
#Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1603.07720
The Khintchine recurrence theorem asserts that on 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 under-recurrent 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.