Metric Number Theory - MAT00049M

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  • Department: Mathematics
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2022-23

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

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.

Module will run

Occurrence Teaching period
A Autumn Term 2022-23

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

 

Syllabus

  • Continued fractions and best approximations to real numbers

  • Hurwitz's theorem

  • Continued fractions and quadratic irrationalities

  • Badly approximable numbers

  • 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

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Special assessment rules

None

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 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.

Indicative reading

G H Hardy and E M Wright, The Theory of Numbers, Oxford University Press.