Topics in Number Theory - Additive Combinatorics — NUT  

Instructor: Dr. Antal BALOG

Text: Instructor's 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.