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Upper and lower class results for subsequences of the Champernowne number

Part of the Lecture Notes in Mathematics book series (LNM,volume 1514)

Abstract

We determine upper and lower bounds for partial sums of subsequences of the dyadic Champernowne sequence, which are obtained from completely deterministic selection functions. This complements results by Shiokawa and Uchiyama.

Keywords

  • Lexicographical Order
  • Normal Sequence
  • Iterate Logarithm
  • Bernoulli Shift
  • Selection Sequence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Borel, E.: Les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Math. Palermo 27, (1909), 247–271.

    CrossRef  MATH  Google Scholar 

  2. Champernowne, D.G.: The construction of decimals normal in the scale of ten. J. London Math. Soc. 52, (1933), 254–260.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Copeland, A.H.; Erdös, P.: Note on normal numbers. Bull. Amer. Math. Soc. 52, (1946), 857–860.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Davenport, H.; Erdös, P.: Note on normal decimals. Canad. J. Math. 4, (1952), 58–63.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Feller, W.: An Introduction to Probability Theory and its Applications. Vol. 1, 7th edition, Wiley, New York-London 1962.

    MATH  Google Scholar 

  6. Kamae, T.: Subsequences of normal sequences. Israel J. Math. 16, (1973), 121–149.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Pillai, S.S.: On normal numbers. Proc. Indian Acad. Sci. Sect. A 10, (1939), 13–15.

    MathSciNet  MATH  Google Scholar 

  8. Pillai, S.S.: On normal numbers. Proc. Indian Acad. Sci. Sect. A 12, (1940), 179–184.

    MathSciNet  MATH  Google Scholar 

  9. Postnikov, A.G.: Arithmetic modelling of stochastic processes. Sel. Transl. in Statistics and Probability 13, (1973), 41–122.

    MATH  Google Scholar 

  10. Shiokawa, I.; Uchiyama, S.: On some properties of the dyadic Champernowne numbers. Acta Math. Acad. Sci. Hun. 26, (1975), 9–27.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Weiss, B.: Normal sequences as collectives. Proc. Symp. on Topological Dynamics and Ergodic Theory, Univ. of Kentucky, 1971.

    Google Scholar 

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© 1992 Springer-Verlag

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Denker, M., Krämer, K.F. (1992). Upper and lower class results for subsequences of the Champernowne number. In: Krengel, U., Richter, K., Warstat, V. (eds) Ergodic Theory and Related Topics III. Lecture Notes in Mathematics, vol 1514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097529

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  • DOI: https://doi.org/10.1007/BFb0097529

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55444-8

  • Online ISBN: 978-3-540-47076-2

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