 

Doug Hensley
Metric Diophantine Approximation and Probability


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


Abstract
Let p_{n}/q_{n}=(p_{n}/q_{n})(x) denote the n^{th} simple continued
fraction convergent to an arbitrary irrational number x∈ (0,1).
Define the sequence of approximation constants
θ_{n}(x):=q_{n}^{2}xp_{n}/q_{n}.
It was conjectured by Lenstra that for almost all x∈(0,1),
lim_{n➜∞}(1/n){j:1≦ j≦ n and
θ_{j}(x)≦ z}=F(z)
where
F(z) :=
z/log 2 if 0≦ z≦ 1/2, and
(1/log 2)(1z+log(2z)) if 1/2≦ z≦ 1.
This was proved in [BJW83] and extended in [Nai98] to the same
conclusion for θ_{kj}(x) where
k_{j} 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 k_{j} be strictly increasing.


Author information
Department of Mathematics, Texas A&M University, College Station, TX 77843
Doug.Hensley@math.tamu.edu
http://www.math.tamu.edu/~doug.hensley/

