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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the limit points of $(a_n\xi )_{n=1}^{\infty }$ mod $1$ for slowly increasing integer sequences $(a_n)_{n=1}^{\infty }$
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by Artūras Dubickas PDF
Proc. Amer. Math. Soc. 137 (2009), 449-456 Request permission

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
  • Artūras Dubickas
  • Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
  • Email: arturas.dubickas@mif.vu.lt
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2448563