# Metrical Theory of Continued Fractions

• Marius Iosifescu
• Cor Kraaikamp
Book

Part of the Mathematics and Its Applications book series (MAIA, volume 547)

1. Front Matter
Pages i-xix
2. Marius Iosifescu, Cor Kraaikamp
Pages 1-51
3. Marius Iosifescu, Cor Kraaikamp
Pages 53-163
4. Marius Iosifescu, Cor Kraaikamp
Pages 165-217
5. Marius Iosifescu, Cor Kraaikamp
Pages 219-311
6. Back Matter
Pages 313-383

### Introduction

This monograph is intended to be a complete treatment of the metrical the­ ory of the (regular) continued fraction expansion and related representations of real numbers. We have attempted to give the best possible results known so far, with proofs which are the simplest and most direct. The book has had a long gestation period because we first decided to write it in March 1994. This gave us the possibility of essentially improving the initial versions of many parts of it. Even if the two authors are different in style and approach, every effort has been made to hide the differences. Let 0 denote the set of irrationals in I = [0,1]. Define the (reg­ ular) continued fraction transformation T by T (w) = fractional part of n 1/w, w E O. Write T for the nth iterate of T, n E N = {O, 1, ... }, n 1 with TO = identity map. The positive integers an(w) = al(T - (W)), n E N+ = {1,2··· }, where al(w) = integer part of 1/w, w E 0, are called the (regular continued fraction) digits of w. Writing . for arbitrary indeterminates Xi, 1 :::; i :::; n, we have w = lim [al(w),··· , an(w)], w E 0, n--->oo thus explaining the name of T. The above equation will be also written as w = lim [al(w), a2(w),···], w E O.

### Keywords

Ergodic theory Probability theory Random variable Stochastic processes continued fraction mixing number theory stochastic process

#### Authors and affiliations

• Marius Iosifescu
• 1
• Cor Kraaikamp
• 2
1. 1.Centre for Math. Statistics “Gheorghe Mihoc”Romanian AcademyBucharestRomania
2. 2.Delft University of Technology, ITS (CROSS)DelftThe Netherlands

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-015-9940-5
• Publisher Name Springer, Dordrecht
• eBook Packages
• Print ISBN 978-90-481-6130-0
• Online ISBN 978-94-015-9940-5
• Buy this book on publisher's site