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.
Arvustused
There are a modest number of exercises at the end of each chapter; most of these are to work out specific numerical examples. I view this as a monograph on a very specialized subject rather than a textbook. (Allen Stenger, MAA Reviews, October 30, 2022)
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.
Oleg Karpenkov is a mathematician at the University of Liverpool (UK), working in the general area of discrete geometry and its applications. More specifically, his research interests include geometry of numbers, discrete and semi-discrete differential geometry and self-stressed configurations of graphs. Oleg 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 in 2012. In 2013 he published a book "Geometry of Continued Fractions" (its extended second edition will be available soon). Currently his Erdos number is 3.
