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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On sequences $(a_n \xi )_{n \ge 1}$ converging modulo $1$
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by Yann Bugeaud PDF
Proc. Amer. Math. Soc. 137 (2009), 2609-2612 Request permission


We prove that, for any sequence of positive real numbers $(g_n)_{n \ge 1}$ satisfying $g_n \ge 1$ for $n \ge 1$ and $\lim _{n \to + \infty } g_n = + \infty$, for any real number $\theta$ in $[0, 1]$ and any irrational real number $\xi$, there exists an increasing sequence of positive integers $(a_n)_{n \ge 1}$ satisfying $a_n \le n g_n$ for $n \ge 1$ and such that the sequence of fractional parts $(\{a_n \xi \})_{n \ge 1}$ tends to $\theta$ as $n$ tends to infinity. This result is best possible in the sense that the condition $\lim _{n \to + \infty } g_n = + \infty$ cannot be weakened, as recently proved by Dubickas.
  • A. Dubickas, On the limit points of $(a_n \xi )_{n=1}^{\infty }$ mod $1$ for slowly increasing integer sequences $(a_n)_{n=1}^{\infty }$, Proc. Amer. Math. Soc. 137 (2009), 449–456.
  • G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
  • Oto Strauch and Štefan Porubský, Distribution of sequences: a sampler, Schriftenreihe der Slowakischen Akademie der Wissenschaften [Series of the Slovak Academy of Sciences], vol. 1, Peter Lang, Frankfurt am Main, 2005. MR 2290224
  • Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864
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Additional Information
  • Yann Bugeaud
  • Affiliation: U.F.R. de Mathématiques, Université Louis Pasteur, 7, rue René Descartes, 67084 Strasbourg, France
  • Email:
  • Received by editor(s): October 6, 2008
  • Received by editor(s) in revised form: November 5, 2008
  • Published electronically: February 4, 2009
  • Communicated by: Ken Ono
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 2609-2612
  • MSC (2000): Primary 11J71, 11K06
  • DOI:
  • MathSciNet review: 2497472