Mastermath course -  Fall 2021

Dr. Jan-Hendrik Evertse
email evertse at math.leidenuniv.nl

Assistants (responsible for the exercise classes and for grading the homework):
Onno Berrevoets  email o.b.berrevoets at math.leidenuniv.nl
Alex Braat  email a.braat at math.leidenuniv.nl

EXAMINATION The examination consists of weekly homework assignments and a written exam. The homework grade is computed by taking the average of the 10 best grades of the homework assignments. The homework exercises will be chosen from the exercises in the lecture notes.
The final grade for Diophantine approximation is computed by taking 25% of the homework grade and 75% of the grade for the written exam, and then rounding off to the closest integer. But to pass the exam, i.e.,to get a grade of 6 or higher, your grade for the written exam has to be at least 5.

The written exam has been scheduled on
Thursday February 3, 2022, 14:00-17:00.

The retake will be either written or oral, depending on the number of students that want to take part. It has been scheduled on
Thursday March 10, 2022, 14:00-17:00.

OLD EXAMS (pdf) Exam 2021/22     Answers to the exercises
Exam 2019/20     Answers to the exercises
Exam 2017/18     Answers to the exercises
Chapter 1: Introduction
Chapter 2: Geometry of numbers
Chapter 3: Algebraic numbers and algebraic number fields
Chapter 4: Transcendence results
Chapter 5: Linear forms in logarithms
Chapter 6: Approximation to algebraic numbers by rationals
Chapter 7: The Subspace Theorem
Chapter 8: The p-adic Subspace Theorem
TIME/PLACE Time:  Thursdays September 17-December 23, 14:00-16:45
Place:  VU, Nieuw Universiteitsgebouw, NU 3A-65, de Boelelaan 1111, Amsterdam
PREREQUISITES Linear algebra;
basic algebra (Groups, rings, fields).
In our course we also need a modest amount of theory of extension of fields and Galois theory over Q. Knowledge of this is convenient, but not necessary, since what we need is in Chapter 3 of the lecture notes.
REMARKS This course will not be given in 2022/23.
LITERATURE The following books are not compulsary, but recommended for further reading:
  • A. Baker, Transcendental Number Theory, Cambridge University Press, 1975.
    Gives a broad but very concise introduction to Diophantine approximation. In particular, the book discusses linear forms in logarithms of algebraic numbers.
    ISBN 0-521-20461-5

  • E.B. Burger, R. Tubbs, Making transcendence transparent, Springer Verlag, 2004.
    Gives a relaxed introduction to transcendence theory.
    ISBN 0-387-21444-5

  • J.W.S. Cassels, An Introduction to the Geometry of Numbers, Springer Verlag, 1997 (reprint of the 1971 edition)
    This book gives a broad introduction to the geometry of numbers.
    ISBN 3-540-61788-4

  • W.M. Schmidt, Diophantine Approximation, Springer Verlag, Lecture Notes in Mathematics 785, 1980.
    This book discusses among other things some basics of geometry of numbers, Roth's Theorem on the approximation of algebraic numbers by rational numbers, Schmidt's own Subspace Theorem, and several applications of the latter.
    ISBN 3-540-09762-7

  • C.L. Siegel, Lectures on the Geometry of Numbers, Springer Verlag, 1989.
    This book contains lecture notes of a course of Siegel on the geometry of numbers, given in 1945/46 in New York. The main topics are a proof of Minkowski's 2nd convex body theorem, and a proof of Kronecker's approximation theorem.
    ISBN 3-540-50629-2
    (tentative, may be subject to change)
    Diophantine approximation deals with problems such as whether a given number is rational/irrational, algebraic/transcendental and more generally how well a given number can be approximated by rational numbers or algebraic numbers. Techniques from Diophantine approximation have been vastly generalized, and today they have many applications to Diophantine equations, Diophantine inequalities, and Diophantine geometry. Our present plan is to discuss the following topics. But this may be subject to changes.

    Geometry of numbers and applications to Diophantine inequalities. Geometry of numbers is concerned with the study of lattice points (points in ℤn) lying in certain bodies in ℝn. We will discuss the two Minkowski's convex bodies theorems. A set C⊂ℝn is called convex if for any two points in C, the line segment connecting these two points is also in C. A closed symmetric convex body in ℝn is a closed, bounded convex set in ℝn which is symmetric about the origin and has the origin as an interior point. Minkowski's first convex body theorem states that if a closed symmetric convex body C⊂ℝn has volume V(C)≥ 2n, then C contains at least one non-zero lattice point.
    Minkowski's second convex body theorem, which is a generalization of the first, deals with the successive minima of a closed, symmetric convex body C⊂ℝn. For λ>0, let λC denote the body obtained by multiplying all points in C by λ. Then the i-th minimum λi of C is the smallest positive λ such that λC contains i linearly independent lattice points from ℤn. Thus a closed, symmetric convex body C⊂ℝn has n successive minima λ1≤...≤ λn. Now Minkowski's second convex body theorem states that the product of the n successive minima of C is about the inverse of the volume of C, more precisely, (2n/n!)V(C)-1≤λ1... λn≤2n V(C)-1.

    Transcendence. We discuss among others the results of Hermite and Lindemann on the transcendence of e and π.

    Approximation of algebraic numbers by rationals. A well-known theorem (which can be deduced for instance using Dirichlet's box principle or Minkowski's convex body theorem but which was known before) asserts that for every real, irrational number α there are infinitely many pairs of integers (x,y) with y>0 and |α-(x/y)|≤ y-2.
    In 1955, K. Roth proved a famous result, stating that if α is an algebraic number, then the exponent -2 on y in the above theorem cannot be replaced by something smaller, more precisely:

    Let α be a real, irrational algebraic number. Then for every d>0 there are only finitely many pairs of integers (x,y) with y>0 and |α-(x/y)|≤ y-2-d.

    Roth's theorem was a culmination of earlier work by Liouville, Thue, Siegel, Gel'fond and Dyson.
    We also intend to discuss a higher dimensional generalization of Roth's Theorem, the so-called Subspace Theorem by W.M. Schmidt (1972), which deals among other things with the simultaneous approximation of algebraic numbers by rationals. This Subspace Theorem is an extremely powerful tool in Diophantine approximation, with many applications to Diophantine equations, linear recurrence sequences, etc.

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