Renewal-type limit theorem for the Gauss map and continued fractions

In this paper we prove a renewal-type limit theorem. Given $\alpha \in (0,1)\backslash \mathbb {Q}$ and R>0, let qnR be the first denominator of the convergents of α which exceeds R. The main result in the paper is that the ratio qnR/R has a limiting distribution as R tends to infinity. The exist...

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Published in:Ergodic theory and dynamical systems 2008-04, Vol.28 (2), p.643-655
Main Authors: SINAI, YAKOV G., ULCIGRAI, CORINNA
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description In this paper we prove a renewal-type limit theorem. Given $\alpha \in (0,1)\backslash \mathbb {Q}$ and R>0, let qnR be the first denominator of the convergents of α which exceeds R. The main result in the paper is that the ratio qnR/R has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map.
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title Renewal-type limit theorem for the Gauss map and continued fractions
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