Some metrical observations on the approximation by continued fractions

Abstract

The following conjecture of H.W. Lenstra is proved. Denote by p n / q n , n = 1,2,… the sequence of continued fraction convergents of the irrational number x and define θ n ( x): = q n | q n x- p n |. Then for every z, 0≤ z≤1, one has for almost all x ▪ Similar results are proved for other functions connected with the regular continued fraction expansion, such as the quotient of |x− p n−1 q n−1 | and |x− p n q n |, as well as for other type of expansions, such as the nearest integer and singular continued fractions. The main tool is the natural extension of the operator x ↦ (1/ x) − [(1/ x)], recently studied by Hitoshi Nakada.