MAGIC085: Metric Number Theory

Course details

A specialist MAGIC course

Semester

Autumn 2022
Monday, October 3rd to Friday, December 9th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Description

Summary

The course is an introduction to the theory of metric Diophantine approximation. This broad and topical area of number theory combines ideas from measure theory, fractal geometry, probability theory, ergodic theory and dynamical systems.

Even at the introductory level, the theory of metric Diophantine approximation naturally illustrates the interplay of different branches of mathematics.

A particular goal of the course is to bring to the forefront the classical and recent `transference' principles that `link' various aspects of the general theory.

For example, the classical Khintchine transference principle provides a link between the homogeneous and inhomogeneous theories.

On the other hand, the recent Mass Transference Principle provides a link between the Lebesgue and Hausdorff measure theories.

Another key goal is to discuss current topical areas of research.

This will involve discussing the fundamental conjectures of Littlewood and Schmidt in the theory of simultaneous Diophantine approximation. 

Useful texts:

  •  G.H. Hardy and E.M. Wright, The theory of numbers, Oxford University Press 
  •  J.W.S. Cassels, An introduction to Diophantine approximation, Cambridge University Press 
  •  W.J. LeVeque: Elementary Theory of Numbers, Addison Wesley Longman Publishing Co. 
  •  W.M. Schmidt: Diophantine Approximation, Springer Verlag. 
  •  Y. Bugeaud: Approximation by algebraic numbers, Cambridge University Press. 
  •  G. Harman: Metric Number Theory, Oxford University Press. 
  •  K. Falconer: Fractal Geometry, John Wiley and Sons Ltd. 
  •  P. Mattila: Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press. 
  •  J. Heinonen: Lectures on analysis on metric spaces, Springer Verlag. 

Prerequisites

  •  A basic 2nd/3rd year course in Elementary Number Theory 
  •  A basic course in Analysis 
  •  A course in measure theory is not essential. 

Syllabus

Topics from:
  •  Dirichlet's theorem in one and higher dimensions. 
  •  Minkowski's theorem - the Geometry of Numbers. 
  •  Inhomogeneous Diophantine approximation. 
  •  Continued fractions and the Gauss map. 
  •  Badly and well approximable numbers. 
  •  Khintchine's Theorem - the Lebesgue measure theory. 
  •  Removing monotonicity - the Duffin-Schaeffer conjecture. 
  •  Jarnik's Theorem - the Hausdorff measure theory. 
  •  General frameworks - ubiquity, regular systems, Schmidt games 
  •  Transference Principles. 
  •  Intersecting Diophantine sets with manifolds and fractals. 
  •  Rational points near manifolds. 

Lecturer

  • SV

    Professor Sanju Velani

    University
    University of York

Bibliography

No bibliography has been specified for this course.

Assessment

The assessment for this course will be released on Monday 9th January 2023 and is due in by Saturday 21st January 2023 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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