Multiple Recurrence and Algorithmic Randomness

TL;DR

This paper explores the relationship between various levels of algorithmic randomness and the validity of effective multiple recurrence theorems in ergodic theory, identifying the minimal randomness needed for such theorems to hold.

Contribution

It determines the specific randomness notions sufficient for effective multiple recurrence theorems in Cantor spaces with different degrees of set effectiveness.

Findings

Kurtz randomness suffices for recurrence into clopen sets.

Schnorr randomness suffices for recurrence into PPI sets with computable measure.

ML-randomness suffices for recurrence into PPI sets.

Abstract

This work contributes to the programme of studying effective versions of "almost everywhere" theorems in analysis and ergodic theory via algorithmic randomness. We determine the level of randomness needed for a point in a Cantor space $ \{0,1\}^{\NN}$ with the uniform measure and the usual shift so that effective versions of the multiple recurrence theorem of Furstenberg holds for iterations starting at the point. We consider recurrence into closed sets that possess various degrees of effectiveness: clopen, $\PPI$ with computable measure, and $\PPI$. The notions of Kurtz, Schnorr, and \ML\ randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the $k$ commuting shift operators on $\{0,1\}^{\NN^{\normalsize k}}$.