Geometry of continued fractions Oleg Karpenkov

Format:

Book

Author/Creator:

Karpenkov, Oleg, author.

Series:

Algorithms and computation in mathematics volume 26

Algorithms and computation in mathematics 1431-1550 volume 26

Language:

English

Subjects (All):

Continued fractions.

Physical Description:

1 online resource

Place of Publication:

Heidelberg Springer [2013]

Language Note:

English

System Details:

text file

PDF

Summary:

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

Contents:

Introduction

Part 1. Regular continued fractions. Classical notions and definitions On integer geometry Geometry of regular continued fractions Complete invariant of integer angles Integer trigonometry for integer angles Integer angles of integer triangles Continued fractions and SL(2; Z) conjugacy classes. Elements of Gauss Reduction Theory. Markoff spectrum Lagrange theorem Gauss-Kuzmin statistics Geometric approximation aspects Geometry of continued fractions with real elements and the second Kepler law Integer angles of polygons and global relations to toric singularities

Part 2. Klein polyhedra. Basic notions and definitions of multidimensional integer geometry On empty simplices, pyramids, parallelepipeds Multidimensional continued fractions in the sense of Klein Dirichlet groups and lattice reduction Periodicity of Klein polyhedra. Generalization of Lagrange theorem Multidimensional Gauss-Kuzmin statistics On construction of multidimensional continued fractions Gauss Reduction in higher dimensions Decomposable forms. Relation to Littlewood and Oppenheim conjectures Approximation of maximal commutative subgroups Other generalizations of continued fractions

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed September 25, 2013)

Other Format:

Printed edition:

ISBN:

9783642393686

3642393683

3642393675

9783642393679

9783642444241

3642444245

9783642393693

3642393691

OCLC:

858924380

Access Restriction:

Restricted for use by site license