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.
References
-
Kargaev P.P., Zhigljavsky A.A. (1996) Approximation of real numbers by
rationals: some metric theorems, Journal of Number Theory, 61, No 2, 209-225.
-
Kargaev P., Zhigljavsky A. (1997) Asymptotic distribution of
the distance function to the Farey points, Journal of Number Theory
65, 130-149.
-
Zhigljavsky A.A., Aliev I. (1999) Weyl sequences: asymptotic distributions
of the partition lengths, Acta Arithmetica, LXXXVIII.4, 351-361.
-
Pronzato L., Wynn H.P., Zhigljavsky A.A. (2001) Analysis of performance
of symmetric second-order line search algorithms through continued fractions,
IMA J of Math Control and Information, 18, 281-296.
-
Huxley M.N. and Zhigljavsky A.A. (2001) On the distribution of Farey fractions
and hyperbolic lattice points, Periodica Matematica Hungarica, 42, 191-198.
-
Moshchevitin N. and Zhigljavsky A.A. (2004) Entropies of the partitions
of the unit interval generated by the Farey tree, Acta Arithmetica, vol. 115,
47-58.
Return
to Personal Home Page
Education | Experience |
Research interests | Publications