Asymptotic Distributions in Diophantine Approximations

    Several interesting features of the accuracy of diophantine approximations can be expressed in probabilistic terms. Many diophantine approximation algorithms produce a sequence of sets F(n), indexed by n, of rational numbers p/q in [0,1]. Famous examples of F(n) are the Farey sequence, the collection of  rationals p/q in [0,1] with q<=n, and the collection of all n-th continued fraction convergents.
    Let F(n) be such a collection of rational numbers and denote d(x,n)=min|x-p/q|, r(x,n)=min q|x-p/q|,where x is in [0,1] and the minima are taken over all rationals p/q in f(n). The quantities d(x,n) and r(x,n) can be considered as distances between x and F(n) and are used to define standard accuracy characteristics of the corresponding diophantine approximation scheme. We also consider several related accuracy characteristics of F(n) such as the average length of the uncertainty interval for a random x in [0,1], Renyi entropies of the partitions generated by the points in F(n), and the two-dimensional distribution of the distances between random x and both endpoints of the uncertainty interval. Properly normalised, these characteristics are expected to exhibit certain asympotic behaviour when n tends to infinity. The corresponding results can typically be expressed in probabilistic language, in terms of weak convergence of sequences of probability measures if x is treated as a random point uniformly distributed on [0,1].
    Studying the limiting probability distributions and the rate of convergence to them constitutes the main objective of the work in this direction.

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