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Transactions of the American Mathematical Society

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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] H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313. Also in Selecta Hermann Weyl, p. 111. MR 1511862
  • [2] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York, and Springer-Verlag, Berlin and New York, 1963. MR 28 #158. MR 551496 (81k:43001)

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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

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