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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A diophantine problem on groups. I


Author: R. C. Baker
Journal: Trans. Amer. Math. Soc. 150 (1970), 499-506
MSC: Primary 42.51
MathSciNet review: 0262774
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Abstract: The following theorem of H. Weyl is generalised to the context of locally compact abelian groups.

Theorem. Let $ {\lambda _1} < {\lambda _2} < {\lambda _3} \cdots $ be a sequence such that, for some $ c > 0,\varepsilon > 0,{\lambda _{n + k}} - {\lambda _n} \geqq c$ whenever $ k \geqq n/{(\log n)^{1 + \varepsilon }}(n = 1,2, \ldots )$. Then for almost all real $ u$ the sequence $ {\lambda _1}u,{\lambda _2}u, \ldots ,{\lambda _n}u\pmod 1$ is uniformly distributed.


References [Enhancements On Off] (What's this?)

  • [1] Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, 10.1007/BF01475864
  • [2] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0262774-8
Keywords: Locally compact abelian groups, sequences of characters, uniform distribution, Weyl's criterion, Haar measure, characters of finite order
Article copyright: © Copyright 1970 American Mathematical Society