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

DOI:
https://doi.org/10.1090/S0002-9939-09-09822-0

Published electronically:
February 4, 2009

MathSciNet review:
2497472

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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.

- 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 https://doi.org/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

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.