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On some properties of the dyadic Champernowne numbers

On some properties of the dyadic Champernowne numbers Acta NIathematiea Academiae Scientiarum Hungarieae Tomus 26 (1---2), (1975), 9--27. ON SOME PROPERTIES OF THE DYADIC CHAMPERNOWNE NUMBERS By I. SHIOKAWA (Tokyo) and S. UCHIYAMA (Okayama) Let x be a real number with 0-<_x< 1 and let X ~. gl~293 ... : ~' ~n2 -n n=l be the dyadic development of x, where the e, = e, (x) are 0 or 1 ; this development is unique unless x is a dyadic rational, in which case to ensure uniqueness we agree to write a terminating expansion in the form in which all digits from a certain point on are 0. The number x is said to be normal (to the base 2, or in the scale of 2), if for any positive integer 1 and any fixed sequence A =~6~...6~ of O's and l's of length l we have lira --I N,(A) = 2 -t, where N,(A) denotes the number of the indices i, l<=i<=n, for which eie~+l .., ei+t_z = ~z~ ... b t. in 1909 E. BOREL [1] proved that almost all real numbers (in the sense of Lebesgue). are normal, and exhibited a normal number in a rather complicated way. Simple explicit examples of normal numbers http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

On some properties of the dyadic Champernowne numbers

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895945
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Abstract

Acta NIathematiea Academiae Scientiarum Hungarieae Tomus 26 (1---2), (1975), 9--27. ON SOME PROPERTIES OF THE DYADIC CHAMPERNOWNE NUMBERS By I. SHIOKAWA (Tokyo) and S. UCHIYAMA (Okayama) Let x be a real number with 0-<_x< 1 and let X ~. gl~293 ... : ~' ~n2 -n n=l be the dyadic development of x, where the e, = e, (x) are 0 or 1 ; this development is unique unless x is a dyadic rational, in which case to ensure uniqueness we agree to write a terminating expansion in the form in which all digits from a certain point on are 0. The number x is said to be normal (to the base 2, or in the scale of 2), if for any positive integer 1 and any fixed sequence A =~6~...6~ of O's and l's of length l we have lira --I N,(A) = 2 -t, where N,(A) denotes the number of the indices i, l<=i<=n, for which eie~+l .., ei+t_z = ~z~ ... b t. in 1909 E. BOREL [1] proved that almost all real numbers (in the sense of Lebesgue). are normal, and exhibited a normal number in a rather complicated way. Simple explicit examples of normal numbers

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: May 21, 2016

References

  • Les probabilités dénombrables et leurs applications arithmétiques
    Borel, E.
  • The construction of decimals normal in the scale of ten
    Champernowne, D. G.
  • Über einen Satz der Wahrscheinlichkeitsrechnung
    Hinĉin, A. Ja.