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On sequences $ (a_n \xi)_{n \ge 1}$ converging modulo $ 1$


Author: Yann Bugeaud
Journal: Proc. Amer. Math. Soc. 137 (2009), 2609-2612
MSC (2000): Primary 11J71, 11K06
Published electronically: February 4, 2009
MathSciNet review: 2497472
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Abstract: 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.


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

  • 1. 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.
  • 2. 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
  • 3. 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
  • 4. Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, 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: bugeaud@math.u-strasbg.fr

DOI: https://doi.org/10.1090/S0002-9939-09-09822-0
Keywords: Distribution modulo $1$
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.