DIOPHANTINE APPROXIMATION - 8 EC

Assistants (responsible for the exercise classes and for grading the homework):
Sebastian Carrillo Santana (UU)   email s.carrillosantana at uu.nl
Mike Daas (UL)   email m.a.daas at math.leidenuniv.nl

The written exam has been scheduled on
Tuesday January 30, 2024, 14:00-17:00, Utrecht, Victor J.Koningsberger buliding, room COSMOS

The retake will be either written or oral, depending on the number of students that want to take part. It has been scheduled on
Tuesday February 27, 2024, 14:00-17:00.

• Homework assignments and their deadlines of delivery will be posted on the ELO page of Diophantine approximation. Please note that these deadlines are strict.
  Homework exercises will be mostly be selected from the exercises in the course notes.
• You may submit your homework either in (la)tex, which we prefer, or handwritten, but in case you submit it in handwritten form it should be very well readable and have no erasures. The teaching assistants who do the grading have the right to ignore your written answers if they cannot read them. Do not forget to write (very well readable) or type your name, university and student number on your homework.
• Homework should be uploaded on the ELO-page of Diophantine Approximation, in the form of a single pdf file (so in case you want to submit it in handwritten form, you shouldn't make pictures of the separate pages, but you should make one single pdf. There are many pdf-scan apps for smartphones).

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

2020 Mathematics Subject Classification (MSC2020)
Official classification of mathematics subjects. Number theory is classified under no. 11.

Number Theory Web
Website for the number theory community with many useful links.

Online number theory lecture notes
Long list of downloadable lecture notes on various branches of number theory.

MathSciNet,   Zentralblatt
Online mathematical data bases which can be used to find abstracts of mathematical papers, lists of papers of mathematicians, etc. MathSciNet covers the period 1940-... and Zentralblatt 1930-... . MathSciNet is accessible only to subscribers. You may consult it through your department's network if your department has subscribed to it. Zentralblatt is now open access.

MacTutor History of Mathematics archive
An archive with all sorts of facts from the history of mathematics, including biographies of the most important mathematicians.

Math arXiv
Freely accessible mathematical preprints archive; number theory preprints are categorized under NT.

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 λ_i 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≤...≤ λ_n. 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... λ_n≤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.