Karpenkov O.N. Geometry of Continued Fractions

2nd. ed. - Springer, 2022. - 461 p. - (Algorithms and Computation in Mathematics, 26). - ISBN 3662652765.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.Preface to the Second Edition.
Acknowledgements.
Preface to the First Edition.
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.
Quadratic forms and Markov Spectrum.
Geometric Continued Fractions.
Continuant Representation of GL(2, Z ) Matrices.
Semigroup of Reduced Matrices.
Elements of Gauss Reduction Theory.
Lagrange’s Theorem.
Gauss—Kuzmin Statistics.
Geometric Aspects of Approximation.
Geometry of Continued Fractions with Real Elements and Kepler’s Second Law.
Extended Integer Angles and Their Summation.
Integer Angles of Polygons and Global Relations for Toric Singularities.
Multidimensional Continued Fractions
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’s Theorem.
Multidimensional Gauss—Kuzmin Statistics.
On the Construction of Multidimensional Continued Fractions.
Gauss Reduction in Higher Dimensions.
Approximation of Maximal Commutative Subgroups.
Other Generalizations of Continued Fractions.
References
IndexOleg Karpenkov has completed his Ph.D. at Moscow State University under the supervision of Vladimir Arnold in 2005. Further he held several postdoctoral positions in Paris (Fellowship of the Mairie de Paris), Leiden, and Graz (Lise Meitner Fellowship) before arriving in Liverpool (2012).True PDF