Normality to Different Bases

  • Gavin Brown


For a positive integer s(> 1), we say that the real number x is normal to base s or s-normal if the sequence (s n x) n-1 is uniformly distributed modulo one. Some 80 years ago Borel showed that almost all real numbers are normal to all bases. Some 40 years ago Steinhaus asked whether 2-normality coincides with 3-normality and Cassels answered the question in the negative (see [3]).




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  1. 1.
    G.Brown & W.Moran, Schmidt’s conjecture on normality for commuting matrices, Inventions Math, to appear.Google Scholar
  2. 2.
    G.Brown, W.Moran & A.Pollington, Normality to non-integer bases, to appear.Google Scholar
  3. 3.
    J.W.S. Cassels, On a problem of Steinhaus about normal numbers, Colloq. Math. 7, 1959, 95–101.MathSciNetMATHGoogle Scholar
  4. 4.
    W.M.Schmidt, Über die normalität von zahlen zu verschiedenen basen, Acta Math. 7, 1962, 299–301.MATHGoogle Scholar
  5. 4.
    W.M.Schmidt, Normalität bezüglich matrizen, J. für die Riene u. Angewandte Math. 2314/5, 1964, 227–260.CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1995

Authors and Affiliations

  • Gavin Brown
    • 1
  1. 1.The University of AdelaideAdelaideAustralia

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