Lectures: MWF 2-3, Math 105.

Office hours: by appointment.

Homework #1, due on Friday, 17 October 2008

Homework #2, due on Monday, 24 November 2008

The course will provide an introduction to the harmonic-analytic approach to additive number theory. We will discuss a certain group of problems at the interface of analytic and combinatorial number theory, harmonic analysis, combinatorics, ergodic theory, with ideas from other fields of mathematics also mixed in. We will start with basic discrete Fourier analysis and move quickly towards active research areas, including Szemerédi's theorem and the Green-Tao theorem. The topics will include the following:

- Discrete Fourier analysis
- Roth's theorem on 3-term arithmetic progressions
- Behrend's example: a large set of integers with no non-trivial 3-term arithmetic progressions
- Varnavides' theorem
- Weyl's equidistribution theorem and exponential sum estimates
- Freiman's theorem on the structure of sets with small sumsets
- The Balog-Szemerédi-Gowers theorem
- Gowers norms and Gowers uniformity
- Gowers' proof of Szemerédi's theorem for 4-term arithmetic progressions
- The Green-Tao theorem: background and selected topics from the proof

There are no formal prerequisites, but a general familiarity with harmonic analysis, combinatorics or number theory will be useful. (The Green-Tao theorem does rely on difficult correlation estimates from analytic number theory; these will be only stated without proof.) I will try to indicate at least some of the current research directions and open problems.

Your course grade will be based on two problem sets, the first one of which will be due in mid-October and the second in late November. The problem sets and the exact due dates will be announced and posted here at least 2 weeks in advance.

- There is no textbook required.
- K. Soundarajan's lecture notes: very accessible, cover most of the course material.
*Additive Combinatorics*, Terence Tao and Van Vu: a comprehensive reference book.- From harmonic analysis to arithmetic combinatorics, my article in Bull. AMS. 45 (2008), 77-115: a general overview of the area covered by the course and many other related things.
- Many research and expository papers are available on Tim Gowers's web page, Ben Green's web page, Terry Tao's web page, and Terry Tao's blog.
- Ben Green's exposition of Freiman's theorem.
- Vsevolod Lev's exposition of the Balog-Szemeredi-Gowers theorem
- Karsten Chipeniuk's exposition of Ben Green's paper on Roth's theorem in the primes
- Number theory books:
- H. Davenport,
*Multiplicative Number Theory*, 2nd ed. (revised by H. Montgomery), Springer-Verlag 1980. - H. Iwaniec and E. Kowalski,
*Analytic Number Theory*, - H. Montgomery,
*Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis*, AMS, 1994. - M. Nathanson,
*Additive Number Theory, I: The Classical Bases; II: Inverse Problems and the Geometry of Sumsets*, Springer-Verlag, 1996. - M. Nathanson,
*Elementary Methods in Number Theory*, Springer-Verlag, 2000. - R.C. Vaughan,
*The Hardy-Littlewood Method*, 2nd ed., Cambridge Univ. Press, 1997 (note: the 2nd edition is substantially different from the 1st).

- H. Davenport,