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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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.

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


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References

  • [1] Bienvenu, L., A. R. Day, M. Hoyrup, I. Mezhirov, and A. Shen, “A constructive version of Birkhoff’s ergodic theorem for Martin-Löf random points,” Information and Computation, vol. 210 (2012), pp. 21–30.
  • [2] Brattka, V., J. S. Miller, and A. Nies, “Randomness and differentiability,” Transactions of the American Mathematical Society, vol. 368 (2016), pp. 581–605.
  • [3] Downey, R. G., and D. R. Hirschfeldt, Algorithmic Randomness and Complexity, Springer, New York, 2010.
  • [4] Franklin, J. N. Y., N. Greenberg, J. S. Miller, and K. M. Ng, “Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets,” Proceedings of the American Mathematical Society, vol. 140 (2012), pp. 3623–28.
  • [5] Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, 1981.
  • [6] Gács, P., M. Hoyrup, and C. Rojas, “Randomness on computable probability spaces—a dynamical point of view,” Theory of Computing Systems, vol. 48 (2011), pp. 465–85.
  • [7] Graham, R. L., B. L. Rothschild, and J. H. Spencer, Ramsey Theory, 2nd edition, Wiley, New York, 1980.
  • [8] Hochman, M., “Upcrossing inequalities for stationary sequences and applications,” Annals of Probability, vol. 37 (2009), pp. 2135–49.
  • [9] Hoyrup, M., “The dimension of ergodic random sequences,” pp. 567–76 in Theoretical Aspects of Computer Science, edited by Christoph Dürr and Thomas Wilke, vol. 14 of Leibniz International Proceedings in Informatics, Schloss Dagstuhl, Wadern, 2012.
  • [10] Kučera, A., “Measure, $\Pi ^{0}_{1}$-classes and complete extensions of $\mathrm{PA}$,” pp. 245–59 in Recursion Theory Week (Oberwolfach, 1984), edited by Heinz-Dieter Ebbinghaus, Gert H. Müller, and Gerald E. Sacks, vol. 1141 of Lecture Notes in Mathematics, Springer, Berlin, 1985.
  • [11] Li, M., and P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, 2nd edition, Springer, New York, 1997.
  • [12] Nies, A., Computability and Randomness, vol. 51 of Oxford Logic Guides, Oxford University Press, Oxford, 2009.
  • [13] Nies, A., editor, Logic Blog 2016 (blog), arxiv.org/abs/1703.01573.
  • [14] Turing, A. M., “On computable numbers, with an application to the Entscheidungsproblem,” Proceedings of the London Mathematical Society, Second Series, vol. 42 (1936), pp. 230–65. Correction, Proceedings of the London Mathematical Society, Second Series, vol. 43 (1937), pp. 544–46.
  • [15] van Lambalgen, M., Random Sequences, Academish Proefschrift, Amsterdam, 1987.
  • [16] Vovk, V. G., “The law of the iterated logarithm for sequences that are random in the sense of Kolmogorov or chaotic (in Russian),” Teoriya Veroyatnosteĭ i ee Primeneniya, vol. 32 (1987), pp. 456–68; English translation in Theory of Probability and Its Applications, vol. 32 (1987), pp. 413–25.
  • [17] V’yugin, V. V., “Effective convergence in probability and an ergodic theorem for individual random sequences (in Russian),” Teoriya Veroyatnosteĭ i ee Primeneniya, vol. 42 (1997), pp. 35–50; English translation in Theory of Probability and Its Applications, vol. 42 (1997), pp. 39–50.