Mathematische Annalen

, Volume 364, Issue 1–2, pp 125–150 | Cite as

On simply normal numbers to different bases

  • Verónica Becher
  • Yann Bugeaud
  • Theodore A. Slaman


Let \(s\) be an integer greater than or equal to \(2\). A real number is simply normal to base \(s\) if in its base-\(s\) expansion every digit \(0, 1, \ldots , s-1\) occurs with the same frequency \(1/s\). Let \({\mathcal{S}}\) be the set of positive integers that are not perfect powers, hence \(\mathcal{S}\) is the set \(\{2,3, 5,6,7,10,11,\ldots \} \). Let \(M\) be a function from \(\mathcal{S}\) to sets of positive integers such that, for each \(s\) in \(\mathcal{S}\), if \(m\) is in \(M(s)\) then each divisor of \(m\) is in \(M(s)\) and if \(M(s)\) is infinite then it is equal to the set of all positive integers. These conditions on \(M\) are necessary for there to be a real number which is simply normal to exactly the bases \(s^m\) such that \(s\) is in \(\mathcal{S}\) and \(m\) is in \(M(s)\). We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases.


  1. 1.
    Becher, V., Slaman, T.A.: On the normality of numbers to different bases. J. Lond. Math. Soc. 90(2), 472–494 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Borel, É.: Les probabilités dénombrables et leurs applications arithmétiques. Supplemento Rendiconti del Circolo Matematico di Palermo 27, 247–271 (1909)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bugeaud, Y.: Distribution Modulo One and Diophantine Approximation. Cambridge Tracts in Mathematics, vol. 193. Cambridge University Press, Cambridge (2012)Google Scholar
  4. 4.
    Bugeaud, Y.: On the expansions of a real number to several integer bases. Revista Matemática Iberoamericana 28(4), 931–946 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Cassels, J.W.S.: On a problem of Steinhaus about normal numbers. Colloq. Math. 7, 95–101 (1959)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Eggleston, H.G.: Sets of fractional dimensions which occur in some problems of number theory. Proc. Lond. Math. Soc. Second Ser. 54, 42–93 (1952)Google Scholar
  7. 7.
    Haiman, M.: Private correspondence (2013)Google Scholar
  8. 8.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)Google Scholar
  9. 9.
    Hertling, P.: Simply normal numbers to different bases. J. Univ. Comput. Sci. 8(2), 235–242 (2002, electronic)Google Scholar
  10. 10.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Dover (2006)Google Scholar
  11. 11.
    Long, C.T.: Note on normal numbers. Pac. J. Math. 7, 1163–1165 (1957)CrossRefzbMATHGoogle Scholar
  12. 12.
    Maxfield, J.E.: Normal \(k\)-tuples. Pac. J. Math. 3, 189–196 (1953)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Pollington, A.D.: The Hausdorff dimension of a set of normal numbers. Pac. J. Math. 95(1), 193–204 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Schmidt, W.M.: On normal numbers. Pac. J. Math. 10, 661–672 (1960)CrossRefzbMATHGoogle Scholar
  15. 15.
    Wolfgang, M.: Schmidt. Über die Normalität von Zahlen zu verschiedenen Basen. Acta Arith. 7, 299–309 (1961/1962)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Verónica Becher
    • 1
  • Yann Bugeaud
    • 2
  • Theodore A. Slaman
    • 3
  1. 1.Universidad de Buenos AiresBuenos AiresArgentina
  2. 2.Université de StrasbourgStrasbourgFrance
  3. 3.University of California BerkeleyBerkeleyUSA

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