Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 581–588 | Cite as

A Theorem About Uniform Distribution

  • Yakov Sinai
Article

Abstract

Many problems of combinatorial number theory can be formulated in terms of behavior of orbits of certain transformations acting on the spaces of integers or their subsets. Their analysis can be reduced to problems embracing number theory, probability theory and dynamical systems. In this paper we consider one such question originated from the famous (3x+1)-problem, which illustrates the difficulties which sometimes arise. The main theorem gives the limiting uniform distribution of certain functionals of independent random variables.

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References

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    Lagarias, J.C.: The 3x+1-problem and its generalizations. Am. Math. Monthly 92, N1, 3–23 (1985)Google Scholar
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    Sinai, Ya.G.: Statistical (3 x + 1) Problem. Commun. in Pure and Applied Math. 56, N7, 1016–1028 (2003)Google Scholar
  3. 3.
    Sinai, Ya.G.: Uniform Distribution in the (3x + 1)-Problem. Moscow Mathematical J. In pressGoogle Scholar
  4. 4.
    Wirsching, G.J.: The dynamical system generated by the 3n+1 function. Lecture Notes in Mathematics 1681, Berlin: Springer-Verlag, 1998Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Yakov Sinai
    • 1
    • 2
  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Landau Institute of Theoretical PhysicsMoscowRussia

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