Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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, fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. MR 568909 (81i:10002)
  • 3. O. Strauch and Š. Porubský, Distribution of sequences: A sampler. Schriftenreihe der Slowakischen Akademie der Wissenschaften, 1. Peter Lang, Frankfurt am Main, 2005. MR 2290224 (2008b:11001)
  • 4. H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313-352. MR 1511862

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11J71, 11K06

Retrieve articles in all journals with MSC (2000): 11J71, 11K06

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.

American Mathematical Society