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 such that the sequence of fractional parts {a u ξ} ∞ n=1 has only finitely many limit points for at least one real irrational number ζ. We prove that, for any sequence of positive numbers (g n ) ∞ n=1 satisfying g n ≥ 1 and lim n→∞ g n = ∞ and any real quadratic algebraic number a, there is an increasing sequence of positive integers (a n ) ∞ n=1 such that a n ≤ n gn for every n ∈ N and lim n→∞ {a n α} = 0. The above bound on an is best possible in the sense that the condition lim n→∞ g n = ∞ cannot be replaced by a weaker condition. More precisely, we show that if (a n ) ∞ n=1 is an increasing sequence of positive integers satisfying lim inf n→∞ a n /n < oo and ζ is a real irrational number, then the sequence of fractional parts {anξ} ∞ n=1 has infinitely many limit points.