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.
Author(s): Oleg Karpenkov (auth.)
Series: Algorithms and Computation in Mathematics 26
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2013
Language: English
Pages: 405
Tags: Algebra; Order, Lattices, Ordered Algebraic Structures; Approximations and Expansions; Convex and Discrete Geometry; Number Theory
Front Matter....Pages I-XVII
Front Matter....Pages 1-2
Classical Notions and Definitions....Pages 3-18
On Integer Geometry....Pages 19-31
Geometry of Regular Continued Fractions....Pages 33-39
Complete Invariant of Integer Angles....Pages 41-46
Integer Trigonometry for Integer Angles....Pages 47-55
Integer Angles of Integer Triangles....Pages 57-65
Continued Fractions and $\operatorname{SL}(2,{\mathbb{Z}})$ Conjugacy Classes. Elements of Gauss’s Reduction Theory. Markov Spectrum....Pages 67-85
Lagrange’s Theorem....Pages 87-97
Gauss–Kuzmin Statistics....Pages 99-114
Geometric Aspects of Approximation....Pages 115-136
Geometry of Continued Fractions with Real Elements and Kepler’s Second Law....Pages 137-151
Extended Integer Angles and Their Summation....Pages 153-172
Integer Angles of Polygons and Global Relations for Toric Singularities....Pages 173-182
Front Matter....Pages 183-184
Basic Notions and Definitions of Multidimensional Integer Geometry....Pages 185-201
On Empty Simplices, Pyramids, Parallelepipeds....Pages 203-214
Multidimensional Continued Fractions in the Sense of Klein....Pages 215-235
Dirichlet Groups and Lattice Reduction....Pages 237-247
Periodicity of Klein Polyhedra. Generalization of Lagrange’s Theorem....Pages 249-270
Multidimensional Gauss–Kuzmin Statistics....Pages 271-280
On Construction of Multidimensional Continued Fractions....Pages 281-300
Front Matter....Pages 183-184
Gauss Reduction in Higher Dimensions....Pages 301-346
Approximation of Maximal Commutative Subgroups....Pages 347-356
Other Generalizations of Continued Fractions....Pages 357-389
Back Matter....Pages 391-405
