Martin-Löf Randomness Implies Multiple Recurrence in Effectively Closed Sets.

This work contributes to the program of studying effective versions of "almost-everywhere" theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space {0, 1}(N) with the uniform measure and the usual shift (erasing the first bit). We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets P of positive measure holds for iterations starting at the point. This means that for each k is an element of N there is an n such that n, 2n, ..., kn shifts of the point all end up in P. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, Pi(0)(1) with computable measure, and Pi(0)(1). The notions of Kurtz, Schnorr, and Martin-Ldf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the k commuting shift operators on {0, 1}(Nk).