MAGIC085: Metric Number Theory

Course details

A specialist MAGIC course


Autumn 2020
Monday, October 5th to Friday, December 11th


Live lecture hours
Recorded lecture hours
Total advised study hours


14:05 - 14:55 (UK)



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. 


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


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. 


  • SV

    Professor Sanju Velani

    University of York
    Main contact
  • VB

    Professor Victor Beresnevich

    University of York


No bibliography has been specified for this course.


The assessment for this course will be released on Monday 11th January 2021 at 00:00 and is due in before Sunday 24th January 2021 at 23:59.

 The assessment for this course will be via a single take-home paper in January with 2 weeks to complete and submit online. There will be 3-4 questions. A pass will be awarded to any script which is judged to be correct on at least half of the questions set. 

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


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