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

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.