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# Metric Number Theory - MAT00049M

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• Department: Mathematics
• Module co-ordinator: Prof. Victor Beresnevich
• Credit value: 10 credits
• Credit level: M
• Academic year of delivery: 2018-19

• None

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## Module will run

Occurrence Teaching cycle
A Autumn Term 2018-19

## Module aims

• To continue the development of number theory.

• To provide a deeper and more quantitative understanding of the structure of the real numbers through Diophantine approximation.

• To illustrate the interplay of different branches of mathematics by the use of algebra, probability and basic results from the theory of Lebesgue measure and fractal geometry.

## Module learning outcomes

At the end of the module you should be able to:

• Understand a range of ideas and techniques in Diophantine approximation.

• Be familiar with the basic use of algebraic and probabilistic ideas within metric number theory.

• Understand the role of fractals within number theory.

• Understand the interplay between number theory and basic dynamical systems.

## Module content

[Pre-requisite knowledge for MSc students: familiarity with and maturity in handling sets, functions, knowledge of (e.g. first courses in) both discrete and analytic number theory.]

Syllabus

• Continued fractions and best approximations to real numbers

• Hurwitz's theorem

• Continued fractions and quadratic irrationalities

• The Borel-Cantelli Lemma

• Khintchine's theorem on approximations by rational numbers

• Hausdorff measures and dimension

• The middle third Cantor set and its dimension

• Hausdorff dimension and badly approximable numbers

• The Jarnik-Besicovitch theorem

• The Pigeonhole Principle and Dirichlet's theorem in higher dimensions

• Minkowski's theorem for convex bodies and systems of linear forms

## Assessment

Task Length % of module mark
University - closed examination
Metric Number Theory
2 hours 100

None

### Reassessment

Task Length % of module mark
University - closed examination
Metric Number Theory
2 hours 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.