## Description

In the ﬁrst part of this book we study geometry of continued fractions in the plane. As usually happens for low dimensions, this case is the richest, and many in-teresting results from different areas of mathematics show up here. While trying to prove analogous multidimensional statements, several research groups have in-vented completely different multidimensional analogues of continued fractions (see, e.g., Chap. 23). This is due to the fact that many relations and regularities that are similar in the planar case become completely different in higher dimensions. So we start with the classical planar case.

The ﬁrst two chapters are preliminary. Nevertheless, we strongly recommend that the reader look through them to get the necessary notation. We start in Chap. 1 with a brief introduction of the notions and deﬁnitions of the classical theory of continued fractions. Further, in Chap. 2, we shed light on the geometry of the planar integer lattice. We present this geometry from the classical point of view, where one has a set of objects and a group of congruences acting on the set of objects. This approach leads to natural deﬁnitions of various integer invariants, such as integer lengths and integer areas.

In Chap. 3 we present the main geometric construction of continued fractions. It is based on the notion of sails, which are attributes of integer angles in integer geometry.

In the next three chapters we come to the problem of the description of integer angles in terms of integer parameters of their sails. We start in Chap. 4 with a con-struction of a complete invariant of integer angles (it is written in terms of integer characteristics of the corresponding sails). Further, in Chap. 5, we use these invari-ants to deﬁne trigonometric functions for integer angles in integer geometry. These functions have many nice properties similar to Euclidean trigonometric functions, while the others are totally different. For instance, there is an integer analogue of the formula α + β + γ = π for the angles in a triangle in the Euclidean plane, which we prove in Chap. 6.

In Chap. 7 we introduce a notion of geometric continued fractions related to arrangements of pairs of lines passing through the origin in the plane. One can con-sider these arrangements to be invariant lines of certain real two-dimensional matrices. So geometric continued fractions give rise to powerful invariants of conjugacy classes of matrices. In particular, we derive from them Gauss’s reduction theory for SL(2, Z)-matrices. We conclude this chapter with a short discussion of the Markov spectrum.

In Chap. 8 we focus on the classical question related to Lagrange’s theorem on the periodicity of continued fractions for quadratic irrationalities, which is based on the structure of the set of solutions of Pell’s equations. In this chapter we also discuss a few questions related to Dirichlet groups of matrices, in particular a problem of solving the equation Xn = A in GL(n, Z).

We continue in Chap. 9 with a classical question of Gauss–Kuzmin statistics for the distribution of elements of continued fractions for arbitrary integers. After the introduction of the classical approach via ergodic theory, we present a geometric meaning of the Gauss–Kuzmin statistics in terms of cross ratios in projective geom-etry.

Regular continued fractions are very good approximations to real numbers. Since there is a vast number of publications on this subject, we do not attempt to cover all interesting problems on continued fractions related to approximations. In Chap. 10 we discuss two questions related to best approximations of real numbers and of arrangements of lines in the plane.

In the last three chapters of the ﬁrst part we introduce applications of contin-ued fractions to differential and algebraic geometry. In Chap. 11 we introduce an “inﬁnitesimal” version of continued fractions arising as sails of planar curves. We show the relation of these fractions to the motion of a body along this curve satis-fying Kepler’s second law. Further, in Chap. 12, we give a supplementary deﬁnition of extended integer angles, which we use to prove the formula on the sum of integer angles in a triangle. In addition, we introduce a deﬁnition of the sum of integer angles, which is a good candidate for the correct addition operation on continued fractions. Finally, in Chap. 13, we describe the global relations on singular points of toric surfaces via integer tangents of the integer angles of polygons associated to these surfaces.