15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 2 - Zahlentheorie
Montag, 17. September 2001, 15.00, Hörsaal 41

 

Some Optimal Error Bounds in the Metric Theory of f-Expansions and Diophantine Approximations

Lothar Heinrich, Universität Augsburg

 

In the first part we consider partial sums $S_n^{(\alpha)}(f) = (f(\xi_1))^{-1/\alpha}+\cdots + (f(\xi_n))^{-1/\alpha}\,,$ where the positve integers $\xi_1, \xi_2,\ldots$ are defined by the $f-$expansion $f( \xi_1 + f( \xi_2 + \cdots ) \,)$ of a real number $\omega \in (0,1)$ which is chosen according to some probability measure ${\sf P}\,$. Under some regularity conditions imposed on the strictly monotone function $f\,:\,(a,\infty)\mapsto (0,1)$ we prove that the suitably normalized sum $S_n^{(\alpha)}(f)$ has an $\alpha-$stable limit distribution ( $0 < \alpha < 2$) and derive uniform bounds of the approximation error provided ${\sf P}$ possesses a strictly positive, Lipschitz continuous Lebesgue density. Special emphasis is put on the case $f(x) = 1/x$ for which $S_n^{(1/p)}(f)$ coincides with the power sum $\xi_1^p + \cdots + \xi_n^p\,$ $( p > 1/2 )\,$ of the partial quotients $\xi_1(\omega) = [1/\omega]\,$ and $\,\xi_k(\omega) = \xi_1(T^{k-1}\omega)$ for $k \ge 2$ (where $T\omega = 1/\omega - [1/\omega]$) of the continuous fraction expansion $\omega = [\xi_1,\xi_2,...]\,$. In the second part we present large deviation relations (in the sense of H. Cramér) for the sequences $\log q_n(\omega)$ and $- \log \vert \omega - p_n(\omega)/q_n(\omega) \vert\,$, where $q_n\,$ ( resp. $p_n\,$) is the denominator (resp. denumerator) of the $n-$th approximant $p_n/q_n = [\xi_1,...,\xi_n]$ of $\omega \in (0,1)\,$. The proofs of the results are based on a method developed in [1].

[1]	HEINRICH, L. (1996) Mixing properties and central limit theorem for a class of non-identical piecewise monotonic $C^2-$ transformations. Math. Nachr. 182, 185 - 214.

E-Mail:	heinrich@math.uni-augsburg.de
Homepage:	www.math.uni-augsburg.de/stochastik/heinrich

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