Geometry of Continued Fractions (eBook)

For different subjects contributed to this book, the author was awarded a Fellowship of the City of Paris (France), a Lise Meitner Fellowship (Austria) and the Moscow Mathematical Society Prize (Russia).

Preface.- Introduction.- 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. Continued fractions and SL(2; Z) conjugacy classes. Elements of Gauss Reduction Theory. Markoff spectrum.- Chapter 8. Lagrange theorem.- Chapter 9. Gauss-Kuzmin statistics.- Chapter 10. Geometric approximation aspects.- Chapter 11. Geometry of continued fractions with real elements and the second Kepler law.- Chapter 12. Integer angles of polygons and global relations to toric singularities.- Part 2. Klein polyhedra: Chapter 13. Basic notions and definitions of multidimensional integer geometry.- Chapter 14. On empty simplices, pyramids, parallelepipeds.- Chapter 15. Multidimensional continued fractions in the sense of Klein.- Chapter 16. Dirichlet groups and lattice reduction.- Chapter 17. Periodicity of Klein polyhedra. Generalization of Lagrange theorem.- Chapter 18. Multidimensional Gauss-Kuzmin statistics.- Chapter 19. On construction of multidimensional continued fractions.- Chapter 20. Gauss Reduction in higher dimensions.- Chapter 21. Decomposable forms. Relation to Littlewood and Oppenheim conjectures.- Chapter 22. Approximation of maximal commutative subgroups.- Chapter 23. Other generalizations of continued fractions.- Bibliography.