Integer Geometry

In many questions, the geometric approach gives an intuitive visualization that leads to a better understanding of a problem and sometimes even to its solution. In the advanced mathematics we give an interpretation of the elements of continued fractions in terms of integer geometry, with the continued fractions being associated to certain invariants of integer angles. The geometric viewpoint on continued fractions also gives key ideas for generalizing Gauss–Kuzmin statistics to studying multidimensional Gauss’s reduction theory, leading to several results in toric geometry. The notion of geometry in general can be interpreted in many different ways.
In this book we think of a geometry as of a set of objects and a congruence relation, which is normally defined by some group of transformations. For instance, in Euclidean geometry in the plane, we study points, lines, segments, polygons, circles, etc., with the congruence relation being defined by the group of all length-preserving transformations O(2, R) (the orthogonal group). The aim of this course is to introduce basic ideas of integer geometry.