Skip to main content
×
×
Home

Uniform distribution and algorithmic randomness

  • Jeremy Avigad (a1)
Abstract

A seminal theorem due to Weyl [14] states that if (an) is any sequence of distinct integers, then, for almost every x ∈ ℝ, the sequence (anx) is uniformly distributed modulo one. In particular, for almost every x in the unit interval, the sequence (anx) is uniformly distributed modulo one for every computable sequence (an) of distinct integers. Call such an x UD random. Here it is shown that every Schnorr random real is UD random, but there are Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random.

Copyright
References
Hide All
[1]Chazelle, Bernard, The discrepancy method, Cambridge University Press, Cambridge, 2000.
[2]Davenport, H., Erdős, P., and LeVeque, W. J., On Weyl's criterion for uniform distribution, The Michigan Mathematical Journal, vol. 10 (1963), pp. 311314.
[3]Downey, Rodney G. and Hirschfeldt, Denis R., Algorithmic randomness and complexity, Springer, New York, 2010.
[4]Harman, Glyn, Metric number theory, The Clarendon Press, New York, 1998.
[5]Kuipers, L. and Niederreiter, H., Uniform distribution of sequences, Wiley-Interscience, New York, 1974.
[6]Loève, Michel, Probability theory II, fourth ed., Springer, New York, 1978.
[7]Lyons, Russell, The measure of nonnormal sets, Inventiones Mathematical vol. 83 (1986), no. 3, pp. 605616.
[8]Merkle, Wolfgang, The Kolmogorov–Loveland stochastic sequences are not closed under selecting subsequences, this Journal, vol. 68 (2003), no. 4, pp. 13621376.
[9]Nies, André, Computability and randomness, Oxford University Press, Oxford, 2009.
[10]Parreau, François and Queffélec, Martine, M0 measures for the Walsh system, The Journal of Fourier Analysis and Applications, vol. 15 (2009), no. 4, pp. 502514.
[11]Rosenblatt, Joseph M. and Wierdl, Máté, Pointwise ergodic theorems via harmonic analysis, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), Cambridge University Press, Cambridge, 1995, pp. 3151.
[12]Ville, J., Étude critique de la notion de collectif, Gauthier-Villars, Paris, 1939.
[13]Wang, Y., Randomness and complexity, Ph.D. thesis, Heidelberg, 1996.
[14]Weyl, Hermann, Über die Gleichverteilung von Zahlen mod. Eins, Mathematische Annalen, vol. 77 (1916), no. 3, pp. 313352.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 87 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd July 2018. This data will be updated every 24 hours.