To give an introduction to the current state of the art in geometry of continued fractions and to study how classical theorems can be visualized via modern techniques of integer geometry.

(LO1) Determine best approximations to real numbers and to homogeneous decomposable forms.

(LO2) Apply the techniques of geometric continued fractions to quadratic irrationalities (Lagrange’s theorem, Markov spectrum).

(LO3) Apply methods of lattice trigonometry in the study of toric varieties.

(LO4) Compute relative frequencies of faces in multidimensional continued fractions.

(LO5) Apply the methods of multidimensional continued fractions to study properties of algebraic irrationalities of higher degree.

(S1) Problem solving skills

(S2) Numeracy

Introduction to theory of continued fractions: generic continued fractions, approximation properties. Lattice geometry, including Pick’s theorem, integer invariants in terms of group indices.

Lattice trigonometry. Basic relations, classification of integer triangles. Toric surfaces, Ikea problem for toric singularities.

Quadratic irrationalities. Lagrange's theorem on the periodicity of continued fractions, the algorithm of Gauss Reduction.

Basics of ergodic theory. Gauss-Kuzmin theorem on the distribution of elements of continued fractions.

Multidimensional continued fractions in the sense of Klein, White’s theorem, classification of two-dimensional faces. Generalized Lagrange theorem. Multidimensional Gauss-Kuzmin distribution. Farey tessellation.