A convergence problem connected with continued fractions

Abstract

The set Z α := { β | lim n → ∞ | | q n β | | = 0 } {Z_\alpha }: = \{ \beta |{\lim _{n \to \infty }}||{q_n}\beta || = 0\} is considered, where ( q n ) n ∈ N {\left ( {{q_n}} \right )_{n \in {\mathbf {N}}}} is the sequence of best approximation denominators of α \alpha , and it is explicitly determined for α \alpha with bounded continued fraction coefficients.