Geometry of Continued Fractions

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Springer Science & Business Media, Aug 15, 2013 - Mathematics - 405 pages

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.

 

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Contents

Regular Continued Fractions
1
Multidimensional Continued Fractions
183
References
390

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About the author (2013)

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).