Geometry of Continued Fractions /

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, classificat...

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Bibliographic Details
Main Author:	Karpenkov, Oleg N. (Author)
Corporate Author:	SpringerLink (Online service)
Format:	eBook
Language:	English
Published:	Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2022.
Edition:	2nd ed. 2022.
Series:	Algorithms and Computation in Mathematics ; 26
Subjects:	Algebra. Approximation theory. Convex geometry . Discrete geometry. Number theory. Order, Lattices, Ordered Algebraic Structures. Approximations and Expansions. Convex and Discrete Geometry. Number Theory. Electronic books.
Online Access:	Connect to the full text of this electronic book

Table of Contents:

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. Lagrange's theorem
Gauss-Kuzmin statistics
Chapter 14. Geometric aspects of approximation
Chapter 15. Geometry of continued fractions with real elements and Kepler's 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 Lagrange's 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.