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≥1 satisfying g n ≥ 1 for n ≥ 1 and lim n→+∞ g n = +∞, for any real number θ in [0,1] and any irrational real number ξ, there exists an increasing sequence of positive integers (a n ) n≥1 satisfying a n ≤ ng n for n ≥ 1 and such that the sequence of fractional parts ({a n ξ}) n≥1 tends to θ as n tends to infinity. This result is best possible in the sense that the condition lim n→+∞ g n = +∞ cannot be weakened, as recently proved by Dubickas.