# Geometry of Continued Fractions

• Oleg Karpenkov
Book

Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 26)

1. Front Matter
Pages I-XVII
2. ### Regular Continued Fractions

1. Front Matter
Pages 1-2
2. Oleg Karpenkov
Pages 3-18
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3. ### Multidimensional Continued Fractions

1. Front Matter
Pages 183-184
2. Oleg Karpenkov
Pages 185-201
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Pages 203-214
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### Introduction

Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. This book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry.

The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses.

### Keywords

algebraic irrationalities continued fractions generalized continued fractions integer trigonometry units of orders

#### Authors and affiliations

• Oleg Karpenkov
• 1
1. 1.Dept. of Mathematical SciencesThe University of Liverpool Dept. of Mathematical SciencesLiverpoolUnited Kingdom

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-39368-6
• Copyright Information Springer-Verlag Berlin Heidelberg 2013
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-642-39367-9
• Online ISBN 978-3-642-39368-6
• Series Print ISSN 1431-1550