Good approximations and continued fractions

Abstract

Let ( q n ) n {({q_n})_n} be the sequence of best approximation denominators of an irrational number α \alpha . The set of real numbers x x for which q n x → 0 {q_n}x \to 0 ( mod 1 ) (\bmod 1) is studied. It is shown that a number x x belongs to α Z ( mod 1) \alpha \mathbb {Z}(\bmod {\text {1)}} if and only if a simple condition on the speed of the convergence related to an arithmetic property of α \alpha is satisfied. This set is uncountable whenever α \alpha has unbounded partial quotients.