# Michael Fuchs: Metric Theory of Continued Fraction Expansions and Diophantine Approximations

We consider the diophantine approximation problem

in integers with where is a function satisfying suitable assumptions and is randomly chosen in the unit interval (according to the Lebesgues measure). We count solutions of this inequality by defining

and

respectively.

We present how metric theory of continued fraction expansion together with probabilistic results for weakly dependent random variables can be used in order to investigate the asymptotic behaviour of the above sequences of random variables. Especially, we present strong invariance principles and invariance principles in distribution for the sequence and a central limit theorem for the sequence (which solves a conjecture of LeVeque) obtained by that method.

## Bibliography

LeV58
William J. LeVeque.
On the frequency of small fractional parts in certain real sequences.
Trans. Amer. Math. Soc., 87:237-261, 1958.