Une application du théorème ergodique sous-additif à la théorie métrique des fractions continues
✍ Scribed by D Barbolosi
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 320 KB
- Volume
- 66
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
For any irrational x # [0, 1] we denote by p n (x)Âq n (x), n=1, 2, ... the sequence of its continued fraction convergents and define % n (x) :=q n |q n x&p n |. Also let T: [0, 1] Ä [0, 1] be defined by T(0)=0 and T(x)=1Âx&[1Âx] if x{0. For some random variables X 1 , X 2 , ..., which are connected with the regular continued fraction expansion, the subadditive ergodic theorem yields to the existence of a function | satisfying: for all z # R,
for almost every x. Furthermore, for X n =% n b T n and X n =(q n&1 Âq n ) b T n , the functions | are explicitly determined. The above results show that the subadditive ergodic theorem can be useful in the metric theory of continued fraction. 1997 Academic Press 1. INTRODUCTION Pour x # (0, 1)$ :=[0, 1]"Q, le de veloppement en fraction continue re gulieÁ re de x sera note par x=[0; a 1 (x), a 2 (x), ..., a n (x), ...] et la suite des convergents correspondants par ( p n (x)Âq n (x)) n 1 . Afin d'e tudier plus article no.