### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title Geometry of Continued Fractions Code MATH447 Coordinator Dr O Karpenkov Mathematical Sciences O.Karpenkov@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level 7 FHEQ Second Semester 15

### Aims

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.

### Learning Outcomes

(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

### Syllabus

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.

### Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH101 Calculus I 2020-21; MATH102 CALCULUS II 2020-21; MATH103 Introduction to Linear Algebra 2020-21

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
written exam  90    50
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Homework 1    10
Homework 2    10
Homework 3    10
Homework 4    10
Homework 5    10