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On the limit points of $ (a_n\xi)_{n=1}^{\infty}$ mod $ 1$ for slowly increasing integer sequences $ (a_n)_{n=1}^{\infty}$


Author: Arturas Dubickas
Journal: Proc. Amer. Math. Soc. 137 (2009), 449-456
MSC (2000): Primary 11B05, 11B37, 11J71, 11R11
DOI: https://doi.org/10.1090/S0002-9939-08-09491-4
Published electronically: August 4, 2008
MathSciNet review: 2448563
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Abstract: In this paper, we are interested in sequences of positive integers $ (a_n)_{n=1}^{\infty}$ such that the sequence of fractional parts $ \{a_n\xi\}_{n=1}^{\infty}$ has only finitely many limit points for at least one real irrational number $ \xi.$ We prove that, for any sequence of positive numbers $ (g_n)_{n=1}^{\infty}$ satisfying $ g_n \geq 1$ and $ \lim_{n\to \infty} g_n=\infty$ and any real quadratic algebraic number $ \alpha,$ there is an increasing sequence of positive integers $ (a_n)_{n=1}^{\infty}$ such that $ a_n \leq n g_n$ for every $ n \in \mathbb{N}$ and $ \lim_{n\to \infty}\{a_n \alpha\} = 0.$ The above bound on $ a_n$ is best possible in the sense that the condition $ \lim_{n\to \infty} g_n=\infty$ cannot be replaced by a weaker condition. More precisely, we show that if $ (a_n)_{n=1}^{\infty}$ is an increasing sequence of positive integers satisfying $ \liminf_{n\to \infty} a_n/n<\infty$ and $ \xi$ is a real irrational number, then the sequence of fractional parts $ \{a_n \xi\}_{n=1}^{\infty}$ has infinitely many limit points.


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Additional Information

Arturas Dubickas
Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
Email: arturas.dubickas@mif.vu.lt

DOI: https://doi.org/10.1090/S0002-9939-08-09491-4
Keywords: Distribution modulo 1, recurrence sequence, quadratic algebraic number
Received by editor(s): December 17, 2007
Received by editor(s) in revised form: January 19, 2008
Published electronically: August 4, 2008
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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