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

We consider the diophantine approximation problem

$$\left\vert x-\frac{p}{q}\right\vert\leq\frac{f(\log q)}{q^2}\eqno (1)$$

in integers $p,q$ with $q>0$ where $f$ is a function satisfying suitable assumptions and $x$ is randomly chosen in the unit interval (according to the Lebesgues measure). We count solutions of this inequality by defining

$$X_n^{(d)}(x):=\char93 \{\langle p,q\rangle\vert 1\leq q\leq n,(p,q)\leq d, \mbox{p/q is a solution of (1)}\}$$

and

$$X_n(x):=\char93 \{\langle p,q\rangle\vert 1\leq q\leq n, \mbox{p/q is a solution of (1)}\}$$

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 $X_n^{(d)}$ and a central limit theorem for the sequence $X_n$ (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.

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