# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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 $(a_n)_{n=1}^{\infty }$HTML articles powered by AMS MathViewer

by Artūras Dubickas
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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