Geometry of Continued Fractions
Oleg Karpenkov
Language: English
Pages: 405
ISBN: 3642393675
Format: PDF / Kindle (mobi) / ePub
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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the shift in indices. 3.2 Geometric Interpretation of the Elements of Continued Fractions 37 3.2 Geometric Interpretation of the Elements of Continued Fractions Let us first formulate a corollary to Theorem 3.1. Corollary 3.5 Consider α ≥ 1. Let A0 A1 A2 . . . and B0 B1 B2 . . . be the principal parts of the corresponding sails (finite or infinite). Then l (Ai Ai+1 ) = a2i and l (Bi Bi+1 ) = a2i+1 , i = 0, 1, 2, . . . , where α = [a0 ; a1 : a2 : · · · ]. The only exception occurs when α
Regular Angles and Related Markov–Davenport Forms 10.2.2 Integer Arrangements and Their Sizes . . . . . . . . . 10.2.3 Discrepancy Functional and Approximation Model . . . . . 99 . . 100 . . 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 101 102 102 104 105 107 108 108 109 109 110 110 111 . . 112 . . 114 . . . . . . . . . . . . . . . . 115 115 115 120 122 123 124 124 xii Contents 10.2.4 Lagrange Estimates for the Case of Continued Fractions with Bounded Elements . . . .
10.1007/978-3-642-39368-6_8, © Springer-Verlag Berlin Heidelberg 2013 87 88 8 Lagrange’s Theorem In case of A with only real eigenvalues, we are interested in the subgroup of the Dirichlet group Ξ (A) containing all matrices with positive eigenvalues. This group is called the positive Dirichlet group and denoted by Ξ+ (A). Definition 8.1 We say that two Dirichlet groups (or positive Dirichlet groups) Ξ1 and Ξ2 are integer congruent if there exists B ∈ GL(2, Z) such that A → B −1 AB is an
R2 equipped with standard metrics on it. Letting l be an arbitrary straight line in R2 that does not pass through the origin, choose some Euclidean coordinates Ol Xl on it. Denote by FCF 1,l a chart of the manifold FCF 1 that consists of all ordered pairs of straight lines both intersecting l. Let us associate to any point of FCF 1,l (i.e. to a collection of two straight lines) coordinates (xl , yl ), where xl and yl are the coordinates on l for the intersections of l with the first and 112 9
congruences. Finally, we formulate a nice expression for the rotation number of a closed integer broken line with integer vertices not passing through the origin introduced in a recent preprint [79] by A. Higashitani and M. Masuda. Recall that the rotation number about the integer point V of a closed broken line L not passing through V is the degree of the projection of L to the unit circle along the rays with vertex at V . We denote the rotation number by RotV (L). Notice that a rotation number
