New York Journal of Mathematics
Volume 4 (1998) 249-257


Doug Hensley

Metric Diophantine Approximation and Probability

Published: December 4, 1998
Keywords: continued fractions, distribution, random variable
Subject: 11K50 primary, 11A55, 60G50 secondary

Let pn/qn=(pn/qn)(x) denote the nth simple continued fraction convergent to an arbitrary irrational number x∈ (0,1). Define the sequence of approximation constants θn(x):=qn2|x-pn/qn|. It was conjectured by Lenstra that for almost all x∈(0,1),
limn➜∞(1/n)|{j:1≦ j≦ n and θj(x)≦ z}|=F(z)
where F(z) := z/log 2 if 0≦ z≦ 1/2, and (1/log 2)(1-z+log(2z)) if 1/2≦ z≦ 1. This was proved in [BJW83] and extended in [Nai98] to the same conclusion for θkj(x) where kj is a sequence of positive integers satisfying a certain technical condition related to ergodic theory. Our main result is that this condition can be dispensed with; we only need that kj be strictly increasing.

Author information

Department of Mathematics, Texas A&M University, College Station, TX 77843