Abstract
We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every \(\Delta ^0_2\)-degree contains an NCR element.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
A proof of this result can be found in [15].
References
Martin-Löf, P.: The definition of random sequences. Inf. Control 9(6), 602–619 (1966)
Zvonkin, A.K., Levin, L.A.: The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russ. Math. Surv. 25(6), 83–124 (1970)
Nies, A.: Computability and Randomness. Oxford Logic Guides, vol. 51. Oxford University Press, Oxford (2009)
Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, New York (2010)
Levin, L.A.: Uniform tests of randomness. Dokl. Akad. Nauk SSSR 227(1), 33–35 (1976)
Reimann, J., Slaman, T.: Measures and their random reals. Trans. Am. Math. Soc. 367(7), 5081–5097 (2015)
Day, A., Miller, J.: Randomness for non-computable measures. Trans. Am. Math. Soc. 365(7), 3575–3591 (2013)
Montalbán, A.: Beyond the arithmetic. PhD thesis, Cornell University (2005)
Cenzer, D., Clote, P., Smith, R.L., Soare, R.I., Wainer, S.S.: Members of countable \(\Pi _1^0\) classes. Ann. Pure Appl. Logic 31, 145–163 (1986)
Barmpalias, G., Greenberg, N., Montalbán, A., Slaman, T.A.: K-trivials are never continuously random. In: Proceedings Of The 11Th Asian Logic Conference: In Honor of Professor Chong Chitat on His 60th Birthday, pp. 51–58. World Scientific (2012)
Reimann, J., Slaman, T.A.: Unpublished work (2008)
Haken, I.R.: Randomizing reals and the first-order consequences of randoms. PhD thesis, UC Berkeley (2014)
Reimann, J., Slaman, T.A.: Effective randomness for continuous measures. J. Am. Math. Soc. 35(2), 467–512 (2022)
Soare, R.I.: Turing Computability. Springer, Berlin (2016)
Li, M.: Randomness and complexity for continuous measures. PhD thesis, Pennsylvania State University (2020)
Reimann, J.: Effectively closed sets of measures and randomness. Preprint arXiv:0804.2656 (2008)
Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)
Acknowledgements
We would like to thank Ted Slaman for many helpful discussions, and for first observing the relation between the granularity function of a measure and the settling time of a real. This crucial initial insight inspired much of the work presented here.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, M., Reimann, J. Turing degrees and randomness for continuous measures. Arch. Math. Logic 63, 39–59 (2024). https://doi.org/10.1007/s00153-023-00873-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-023-00873-7
Keywords
- Algorithmic randomness
- Continuous measures
- Turing degrees
- Recursively enumerable and above
- Moduli of computation