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arXiv · 0710.1283

Renewal-type Limit Theorem for the Gauss Map and Continued Fractions

Yakov G. Sinai·Corinna Ulcigrai
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Abstract

In this paper we prove the following renewal-type limit theorem. Given an irrational $α$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of the convergents of $α$ which exceeds R. The main result in the paper is that the ratio $q_{n_R}/R$ 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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Yakov G. Sinai, Corinna Ulcigrai. 2007-10-05. Renewal-type Limit Theorem for the Gauss Map and Continued Fractions. https://arxiv.org/abs/0710.1283

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