On sequences $(a_n \xi )_{n \ge 1}$ converging modulo $1$

Abstract:

We prove that, for any sequence of positive real numbers $(g_n)_{n \ge 1}$ satisfying $g_n \ge 1$ for $n \ge 1$ and $\lim _{n \to + \infty } g_n = + \infty$, for any real number $\theta$ in $[0, 1]$ and any irrational real number $\xi$, there exists an increasing sequence of positive integers $(a_n)_{n \ge 1}$ satisfying $a_n \le n g_n$ for $n \ge 1$ and such that the sequence of fractional parts $(\{a_n \xi \})_{n \ge 1}$ tends to $\theta$ as $n$ tends to infinity. This result is best possible in the sense that the condition $\lim _{n \to + \infty } g_n = + \infty$ cannot be weakened, as recently proved by Dubickas.