Notre Dame Journal of Formal Logic

Martin-Löf Randomness Implies Multiple Recurrence in Effectively Closed Sets

Rodney G. Downey, Satyadev Nandakumar, and André Nies

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Primary: 03D32: Algorithmic randomness and dimension [See also 68Q30]
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35}

algorithmic randomness symbolic dynamics mutiple recurrence effectively closed sets


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.

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