DIOPHANTINE APPROXIMATIONCourse for third year bachelor and master students  Fall 2017[ estudiegids Bachelor wiskunde  eprospectus Master mathematics  Homepage Mathematical Institute ] 

EXAM  Exam Answers to the exercises  
TEACHING 
Teacher: Dr. JanHendrik Evertse office 248, tel. 0715277148, email evertse at math.leidenuniv.nl
Assistant (responsible for grading the homework):  
EC POINTS  6 or 8
The basic course is for 6 EC, but interested students may extend this to 8 EC by studying some additional literature and doing some extra homework. You may decide whether you want to extend your number of EC to 8 after you have obtained your grade for 6EC. Note: Bachelor students who want to investigate the possibility of letting the EC of this course count towards their master's diploma, are adivsed to contact the chairman of the Exam Committee (Ronald van Luijk, rvl at math.leidenuniv.nl) at their earliest convenience.  
EXAMINATION FOR 6EC 
The examination consists of four homework assignments,
issued once in approximately three weeks, and a written exam.
The written exam has been scheduled on:
Tuesday January 23, 2018, 14:0017:00, Snellius room 174. To get a sufficient final grade of 6 or higher, the grade for the written exam should be at least 5. Then the final grade for the course (for 6EC) is determined by 0.5x(average of the grades for the four homework assignments)+0.5x(grade for written exam), rounded to the closest half, and rounded up to 6 if it is between 5.5 and 6.
At the written exam questions may be asked about the theorems and proofs of chapters 17 of the lecture notes as well as the exercises of the four homework assignments. We will not ask questions about the following sections and parts:
 
EXTENSION TO 8 EC 
Those of you who want to extend the number
of ECpoints from 6 to 8, should first get a sufficient grade
of 6 or higher for 6EC
and then deliver the extra homework assignment, which will be posted below
under HOMEWORK.
To get the two extra EC, the grade for the extra assignment should
be at least 5.5. Then
the final grade for 8EC is determined by
0.75x(unrounded grade for 6EC)+0.25x(unrounded grade for extra assignment), rounded to the closest half, and rounded up to 6 if it is between 5.5 and 6. You will have to do your written exam before doing this extra assignment. 

COURSE NOTES (pdf) 
Chapter 1: Introduction
Chapter 2: Geometry of numbers (updated September 14) Chapter 3: Algebraic numbers and algebraic number fields Chapter 4: Transcendence results (correction Exercise 4.6 November 28) Chapter 5: Linear forms in logarithms Chapter 6: Approximation to algebraic numbers by rationals Chapter 7: The Subspace Theorem For the two additional EC, you have to study the following two chapters, and submit the extra homework assignment posted below:
Chapter 8: padic numbers
 
HOMEWORK (pdf) 
HOMEWORK GRADES (xlsx)
Assignments and deadlines:
 
TIME/PLACE  Time: Wednesdays September 6December 13, 11:0012:45, with the exception of October 4, November 8.
Place: Snellius, room 402 Further information on the schedules can be found on the Course schedule webpage. 

PREREQUISITES 
Algebra 1, Algebra 2 (Groups, rings, fields).
In our course we also need a modest amount of theory of extension of fields and Galois theory (Algebra 3). Knowledge of this is convenient, but not necessary, since we will recall what is needed during the course.  
REMARKS 
This course will not be given in 2018/2019.
This course is recommended for a Master's thesis project in Number Theory. 

LITERATURE  The following books are not compulsary,
but recommended for further reading:
Gives a broad but very concise introduction to Diophantine approximation. In particular, the book discusses linear forms in logarithms of algebraic numbers. ISBN 0521204615
Gives a relaxed introduction to transcendence theory. ISBN 0387214445
This book gives a broad introduction to the geometry of numbers. ISBN 3540617884
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 3540097627
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 3540506292  
USEFUL WEBSITES 
Official classification of mathematics subjects. The books in the Mathematical Institute library are classified according to this classification. The number theory books are classified under no. 11.
Website for the number theory community with many useful links.
Long list of downloadable lecture notes on various branches of number theory.
Online mathematical data bases. In principle you can access these databases only by logging on in the university network with your ULCN username and password. I am not sure whether this will work for you (for me it works perfectly), but you may try to access these databases remotely by setting up a proxy connection on your own pc or laptop. For instructions, see Computers at the Mathematical Institute→remote access
An archive with all sorts of facts from the history of mathematics, including biographies of the most important mathematicians.
arXiv preprint server with link to the number theory preprints 

CONTENTS (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
ℤ^{n}) lying in certain bodies
in ℝ^{n}.
We will discuss the two
Minkowski's convex bodies theorems.
A set C⊂ℝ^{n} 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 ℝ^{n}
is a closed, bounded convex set in ℝ^{n}
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⊂ℝ^{n} has volume
V(C)≥ 2^{n}, then C contains at least
one nonzero lattice point.
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 wellknown 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}.
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^{2d}.
Roth's theorem was a culmination of earlier work by
Liouville,
Thue,
Siegel,
Gel'fond and
Dyson.

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