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Geometry of continued fractions / Oleg Karpenkov.

Karpenkov, Oleg, author.
2022
QA295
Available Online
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Title
Geometry of continued fractions / Oleg Karpenkov.
Author
Karpenkov, Oleg, author.
Edition
Second edition.
ISBN
9783662652770 (electronic bk.)
3662652773 (electronic bk.)
9783662652763
3662652765
DOI
https://doi.org/10.1007/978-3-662-65277-0
Published
Berlin : Springer, 2022.
Language
English
Description
1 online resource : illustrations (black and white).
Item Number
10.1007/978-3-662-65277-0 doi
Call Number
QA295
Dewey Decimal Classification
515/.243
Summary
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 second edition now includes a geometric approach to Gauss Reduction Theory, classification of integer regular polygons and some further new subjects. 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. 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.
Note
Previous edition: 2013.
Bibliography, etc. Note
Includes bibliographical references and index.
Access Note
Access limited to authorized users.
Source of Description
Description based on print version record.
Series
Algorithms and computation in mathematics ; v. 26.
Available in Other Form
Geometry of continued fractions.
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Part 1. Regular continued fractions: Chapter 1. Classical notions and definitions
Chapter 2. On integer geometry
Chapter 3. Geometry of regular continued fractions
Chapter 4. Complete invariant of integer angles
Chapter 5. Integer trigonometry for integer angles
Chapter 6. Integer angles of integer triangles
Chapter 7. Quadratic forms and Makov spectrum.
Chapter 8. Geometric continued fractions
Chapter 9. Continuant representation of GL(2,Z) Matrices
Chapter 10. Semigroup of Reduced Matrices
Chapter 11. Elements of Gauss reduction theory
Chapter 12. Lagranges theorem
Gauss-Kuzmin statistics
Chapter 14. Geometric aspects of approximation
Chapter 15. Geometry of continued fractions with real elements and Keplers second law
Chapter 16. Extended integer angles and their summation
Chapter 17. Integer angles of polygons and global relations for toric singularities
Part II. Multidimensional continued fractions
Chapter 18. Basic notations and definitions of multidimensional integer geometry
Chapter 19. On empty simplices, pyramids, parallelepipeds
Chapter 20. Multidimensional continued fractions in the sense of Klein
Chapter 21. Dirichlet groups and lattice reduction
Chapter 22. Periodicity of Klein polyhedral. Generalization of Lagranges Theorem
Chapter 23. Multidimensional Gauss-Kuzmin Statistics
Chapter 24. On the construction of multidimensional continued fractions
Chapter 25. Gauss reduction in higher dimensions. Chapter 26. Approximation of maximal commutative subgroups
Capter 27. Other generalizations of continued fractions. References. Index.

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