Notre Dame Journal of Formal Logic

Normal Numbers and Limit Computable Cantor Series

Achilles Beros and Konstantinos Beros

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Given any oracle, $A$, we construct a basic sequence $Q$, computable in the jump of $A$, such that no $A$-computable real is $Q$-distribution-normal. A corollary to this is that there is a $\Delta^{0}_{n+1}$ basic sequence with respect to which no $\Delta^{0}_{n}$ real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.

Article information

Source
Notre Dame J. Formal Logic Volume 58, Number 2 (2017), 215-220.

Dates
Received: 8 April 2014
Accepted: 21 November 2014
First available in Project Euclid: 22 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1490148081

Digital Object Identifier
doi:10.1215/00294527-2017-0004

Subjects
Primary: 03D28: Other Turing degree structures
Secondary: 03D80: Applications of computability and recursion theory

Keywords
computability theory recursion theory Turing degrees number theory normal numbers Cantor series expansions basic series

Citation

Beros, Achilles; Beros, Konstantinos. Normal Numbers and Limit Computable Cantor Series. Notre Dame J. Formal Logic 58 (2017), no. 2, 215--220. doi:10.1215/00294527-2017-0004. https://projecteuclid.org/euclid.ndjfl/1490148081.


Export citation

References

  • [1] Adler, R., M. Keane, and M. Smorodinsky, “A construction of a normal number for the continued fraction expansion,” Journal of Number Theory, vol. 13 (1981), pp. 95–105.
  • [2] Altomare, C., and B. Mance, “Cantor series constructions contrasting two notions of normality,” Monatshefte für Mathematik, vol. 164 (2011), pp. 1–22.
  • [3] Becher, V., P. A. Heiber, and T. A. Slaman, “A polynomial-time algorithm for computing absolutely normal numbers,” Information and Computation, vol. 232 (2013), pp. 1–9.
  • [4] Becher, V., P. A. Heiber, and T. A. Slaman, “Normal numbers in the Borel hierarchy,” Fundamenta Mathematicae, vol. 226 (2014), pp. 63–78.
  • [5] Cantor, G., “Ueber die einfachen Zahlensysteme,” Zeitschrift für Mathematik und Physik, vol. 14 (1869), pp. 121–28.
  • [6] Champernowne, D. G., “The construction of decimals normal in the scale of ten,” Journal of the London Mathematical Society, vol. S1-8 (1933), pp. 254–60.
  • [7] Erdös, P., and A. Rényi, “On Cantor’s series with convergent $\sum1/q_{n}$,” Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, vol. 2 (1959), pp. 93–109.
  • [8] Erdös, P., and A. Rényi, “Some further statistical properties of the digits in Cantor’s series,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 10 (1959), pp. 21–29.
  • [9] Ki, H., and T. Linton, “Normal numbers and subsets of $\mathbb{N}$ with given densities,” Fundamenta Mathematicae, vol. 144 (1994), pp. 163–79.
  • [10] Madritsch, M. G., “Generating normal numbers over Gaussian integers,” Acta Arithmetica, vol. 135 (2008), pp. 63–90.
  • [11] Madritsch, M. G., J. M. Thuswaldner, and R. F. Tichy, “Normality of numbers generated by the values of entire functions,” Journal of Number Theory, vol. 128 (2008), pp. 1127–45.
  • [12] Nies, A., Computability and Randomness, vol. 51 of Oxford Logic Guides, Oxford University Press, Oxford, 2009.

Duke University Press

Notre Dame Journal of Formal Logic

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more