Let $p_n/q_n=(p_n/q_n)(x)$ denote the $n^{\hbox{th}}$ simple continued fraction convergent to an arbitrary irrational number $x\in (0,1)$. Define the sequence of approximation constants $\theta_n(x):=q_n^2|x-p_n/q_n|$. It was conjectured by Lenstra that for almost all $x\in(0,1)$,

\[\lim_{n\rightarrow\infty}\frac{1}{n}|\{j:1\le j\le n\hbox{\ and\ } \theta_j(x)\le z\}|=F(z)\]

where $F(z) := z/\log 2$ if $0\le z\le 1/2$, and $\frac{1}{\log 2}(1-z+\log(2z))$ if $1/2\le z\le 1$. This was proved in [BJW83] and extended in [Nai98] to the same conclusion for $\theta_{k_j}(x)$ 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.

[BJW83]W.~Bosma, H.~Jager, and F.~Wiedijk, {\em Some metrical observations on the approximation by continued fractions}, Indag. Math. {\bf 45} (1983), 281--299, MR 85f:11059, Zbl 519.10043.

[Nai98]R.~Nair, \hyperlink{http://nyjm.albany.edu:8000/j/1998/3A-9.html} {\em On metric diophantine approximation theory and subsequence ergodic theory}, New York Journal of Mathematics {\bf 3A} (1998), 117--124, MR 99b:11088, Zbl 894.11032.