*Instructor:* Dr. Antal BALOG

*Text: *Instructors handout.

*Reference text: *
T. Tao, V. Vu, Additive
Combinatorics, Cambridge University Press, 2006 *or* M. B.
Nathanson, Additive Number Theory, Inverse Problems and the
Geometry of Sumsets, Springer Verlag, 1996.

*Prerequisite:* A first course of number theory; you must be
familiar with prime numbers, divisibility and congruences.
Algebra; you must be familiar with the basic concepts of linear
algebra, and groups. Calculus; you must be familiar with
functions, differential calculus, integrals. Combinatorics; you
must be familiar with the basic concept of graphs, but no graph
theoretic results are used in this course. Actually the course
uses rather elementary tools with almost no prerequisite, still
the arguments are going to be deep and involved sometimes.

*Subject:*
Our aim is to give an introduction to
Additive Combinatorics, one of the most recent and most
dynamically developing branch of Number Theory. We will cover
classical direct problems, such as Roth's Theorem about three
term arithmetic progressions, as well as inverse problems, such
as Freimann's Theorem about the classification of sets with small
doubling. The most spectacular results of the subject,
Szemeredi's Theorem about long arithmetic progressions in dense
sets or Green--Tao's Theorem about long arithmetic progressions
in primes are beyond the scope of a one semester course, but
interested students get the necessary basis to continue their
studies in this interesting field.