## Notre Dame Journal of Formal Logic

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

### Martin-Löf Randomness Implies Multiple Recurrence in Effectively Closed Sets

Rodney G. Downey, Satyadev Nandakumar, and André Nies

#### 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 $\{0,1{\}}^{{\mathbb{N}}^{k}}$.

#### 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

**Subjects**

Primary: 03D32: Algorithmic randomness and dimension [See also 68Q30]

Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35}

**Keywords**

algorithmic randomness symbolic dynamics mutiple recurrence effectively closed sets

#### 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