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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?)

  • 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.
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  • 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
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Additional Information

Yann Bugeaud
Affiliation: U.F.R. de Mathématiques, Université Louis Pasteur, 7, rue René Descartes, 67084 Strasbourg, France

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.