Notre Dame Journal of Formal Logic

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 $n,2n,\ldots ,kn$ shifts of the point all end up in $\mathcal{P}$. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, $\Pi ^{0}_{1}$ with computable measure, and $\Pi ^{0}_{1}$. 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 $\{0,1\}^{{\mathbb{N}}^{k}}$.

Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 3 (2019), 491-502.

Dates
Received: 9 June 2017
Accepted: 17 June 2017
First available in Project Euclid: 11 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1562810592

Digital Object Identifier
doi:10.1215/00294527-2019-0017

Mathematical Reviews number (MathSciNet)
MR3985623

Citation

Downey, Rodney G.; Nandakumar, Satyadev; Nies, André. Martin-Löf Randomness Implies Multiple Recurrence in Effectively Closed Sets. Notre Dame J. Formal Logic 60 (2019), no. 3, 491--502. doi:10.1215/00294527-2019-0017. https://projecteuclid.org/euclid.ndjfl/1562810592

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