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

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.