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General


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.

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.

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.

Semester

Autumn 2016 (Monday, October 3 to Friday, December 9)

Timetable

  • Wed 11:05 - 11:55

Prerequisites

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.

Lecturers


Sanju Velani (main contact)
Email slv3@york.ac.uk
Phone (01904) 324599
vcard
Victor Beresnevich
Email victor.beresnevich@york.ac.uk
Phone
vcard


Students


Photo of Ardavan Afshar
Ardavan Afshar
(*External)
Photo of Philip Carter
Philip Carter
(Liverpool)
Photo of Tanmay Inamdar
Tanmay Inamdar
(East Anglia)
Photo of Oleksiy Klurman
Oleksiy Klurman
(*External)
Photo of Matthew Northey
Matthew Northey
(Durham)
Photo of Matthew Poulter
Matthew Poulter
(Lancaster)
Photo of Sam Povall
Sam Povall
(Liverpool)
Photo of Angelo Rendina
Angelo Rendina
(Sheffield)
Photo of Matty Van Son
Matty Van Son
(Liverpool)
Photo of Jared White
Jared White
(Lancaster)


Bibliography


No bibliography has been specified for this course.

Assessment



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 4 questions and you will need the equivalent of 2 questions to pass.

MAGIC_exam2017

Files:Exam paper
Released: Sunday 8 January 2017 (111.0 days ago)
Deadline: Monday 23 January 2017 (95.0 days ago)


Files


Files marked L are intended to be displayed on the main screen during lectures.

Week(s)File
MAGIC_exam2017.pdf
ProblemsMagic.pdf
0MagicNotes.pdfL


Recorded Lectures


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