Martin-Löf Randomness Implies Multiple Recurrence in Effectively Closed Sets

Abstract

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 $\{0,1{\}}^{\mathbb{N}}$ 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 $\mathcal{P}$ of positive measure holds for iterations starting at the point. This means that for each $k\in \mathbb{N}$ there is an $n$ such that $n,2n,\dots ,kn$ shifts of the point all end up in $\mathcal{P}$. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, ${\Pi }_1^0$ with computable measure, and ${\Pi }_1^0$. The notions of Kurtz, Schnorr, and Martin-Löf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the $k$ commuting shift operators on $\{0,1{\}}^{{\mathbb{N}}^k}$.