Advertisement

The Ramanujan Journal

, Volume 29, Issue 1–3, pp 409–422 | Cite as

Nonnormality of Stoneham constants

  • David H. Bailey
  • Jonathan M. BorweinEmail author
Article

Abstract

This paper examines “Stoneham constants,” namely real numbers of the form \(\alpha_{b,c} = \sum_{n \geq1} 1/(c^{n} b^{c^{n}})\), for coprime integers b≥2 and c≥2. These are of interest because, according to previous studies, α b,c is known to be b-normal, meaning that every m-long string of base-b digits appears in the base-b expansion of the constant with precisely the limiting frequency b m . So, for example, the constant \(\alpha_{2,3} = \sum_{n \geq1} 1/(3^{n} 2^{3^{n}})\) is 2-normal. More recently it was established that α b,c is not bc-normal, so, for example, α 2,3 is provably not 6-normal. In this paper, we extend these findings by showing that α b,c is not B-normal, where B=b p c q r, for integers b and c as above, p,q,r≥1, neither b nor c divide r, and the condition D=c q/p r 1/p /b c−1<1 is satisfied. It is not known whether or not this is a complete catalog of bases to which α b,c is nonnormal. We also show that the sum of two B-nonnormal Stoneham constants as defined above, subject to some restrictions, is B-nonnormal.

Keywords

Normality of irrational numbers Nonnormality of irrational numbers Stoneham numbers Normality of sums 

Mathematics Subject Classification

11K16 11K31 

Notes

Acknowledgements

The authors wish to express their profound appreciation to Boris Adamczewski who, as referee, made a very incisive report. This provided Lemma 2 and sketched proofs of Theorems 2 and 3. Lemma 2 in particular appears to be a very useful additional tool for research into the normality of mathematical constants.

References

  1. 1.
    Adamczewski, B., Bugeaud, Y.: On the complexity of algebraic numbers I. Expansions in integer bases. Ann. Math. 165, 547–565 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Adamczewski, B., Faverjon, C.: Non-zero digits in the expansion of irrational algebraic numbers in an integer base. C. R. Acad. Sci. Paris, Ser. I 350, 1–4 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Allouche, J.-P., Shallit, J.O.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003) zbMATHCrossRefGoogle Scholar
  4. 4.
    Aragon Artacho, F.J., Bailey, D.H., Borwein, J.M., Borwein, P.B.: Tools for visualizing real numbers: planar number walks. Manuscript, 30 May 2012, available at http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/tools-vis.pdf
  5. 5.
    Bailey, D., Borwein, J.: Normal numbers and pseudorandom generators. In: Proceedings of the Workshop on Computational and Analytical Mathematics in Honour of Jonathan Borwein’s 60th Birthday September, Springer (2011, to appear), available at http://crd.lbl.gov/~dhbailey/dhbpapers/normal-pseudo.pdf
  6. 6.
    Bailey, D.H., Crandall, R.E.: Random generators and normal numbers. Exp. Math. 11, 527–546 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bailey, D.H., Misiurewicz, M.: A strong hot spot theorem. Proc. Am. Math. Soc. 134, 2495–2501 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bailey, D.H., Borwein, J.M., Crandall, R.E., Pomerance, C.: On the binary expansions of algebraic numbers. J. Number Theory Bordx. 16, 487–518 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bailey, D.H., Borwein, J.M., Calude, C.S., Dinneen, M.J., Dumitrescu, M., Yee, A.: Normality and the digits of pi. Exp. Math. (2012, to appear), available at http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/normality.pdf
  10. 10.
    Becher, V., Figueira, S.: An example of a computable absolutely normal number. Theor. Comput. Sci. 270, 947–958 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Becher, V., Figueira, S.: Turing’s unpublished algorithm for normal numbers. Theor. Comput. Sci. 377, 126–138 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Berggren, L., Borwein, J.M., Borwein, P.B.: Pi: A Source Book, 3rd edn. Springer, Berlin (2004) zbMATHGoogle Scholar
  13. 13.
    Borwein, J., Bailey, D.H.: Mathematics by Experiment: Plausible Reasoning in the 21st Century. AK Peters, Natick (2008) zbMATHGoogle Scholar
  14. 14.
    Borwein, J.M., Borwein, P.B.: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley-Interscience, New York (1987) zbMATHGoogle Scholar
  15. 15.
    Borwein, J.M., Borwein, P.B.: On the generating function of the integer part: [+γ]. J. Number Theory 43, 293–318 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Calude, C.S.: Borel normality and algorithmic randomness. In: Rozenberg, G., Salomaa, A. (eds.) Developments in Language Theory, pp. 113–119. World Scientific, Singapore (1994) Google Scholar
  17. 17.
    Calude, C.S., Dinneen, M.J.: Computing a glimpse of randomness. Exp. Math. 11, 361–370 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Copeland, A.H., Erdős, P.: Note on normal numbers. Bull. Am. Math. Soc. 52, 857–860 (1946) zbMATHCrossRefGoogle Scholar
  19. 19.
    Hertling, P.: Simply normal numbers to different bases. J. Univers. Comput. Sci. 8, 235–242 (2002) MathSciNetGoogle Scholar
  20. 20.
    Kaneko, H.: On normal numbers and powers of algebraic numbers. Integers 10, 31–64 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Martin, G.: Absolutely abnormal numbers. Am. Math. Mon. 108, 746–754 (2001) zbMATHCrossRefGoogle Scholar
  22. 22.
    Stoneham, R.: On absolute (j,ε)-normality in the rational fractions with applications to normal numbers. Acta Arith. 22, 277–286 (1973) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Lawrence Berkeley National LaboratoryBerkeleyUSA
  2. 2.Centre for Computer Assisted Research Mathematics and its Applications (CARMA)University of NewcastleCallaghanAustralia

Personalised recommendations