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August 2019 Martin-Löf Randomness Implies Multiple Recurrence in Effectively Closed Sets
Rodney G. Downey, Satyadev Nandakumar, André Nies
Notre Dame J. Formal Logic 60(3): 491-502 (August 2019). DOI: 10.1215/00294527-2019-0017

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}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 kN 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, Π10 with computable measure, and Π10. 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}Nk.

Citation

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Rodney G. Downey. Satyadev Nandakumar. André Nies. "Martin-Löf Randomness Implies Multiple Recurrence in Effectively Closed Sets." Notre Dame J. Formal Logic 60 (3) 491 - 502, August 2019. https://doi.org/10.1215/00294527-2019-0017

Information

Received: 9 June 2017; Accepted: 17 June 2017; Published: August 2019
First available in Project Euclid: 11 July 2019

zbMATH: 07120752
MathSciNet: MR3985623
Digital Object Identifier: 10.1215/00294527-2019-0017

Subjects:
Primary: 03D32
Secondary: 37A30

Rights: Copyright © 2019 University of Notre Dame

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Vol.60 • No. 3 • August 2019
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