The course has been rescheduled to meet on Tuesdays and Thursdays 2 - 4, except that we do not meet on those Thursdays on which the Number Theory seminar is scheduled.

The general theme of this course will be "applications of Fourier analysis in number theory". This will include topics in combinatorial and additive number theory (Roth's theorem, Freiman's theorem) as well as more classical analytic number theory (Hardy-Littlewood circle method, Weyl's inequality). I am hoping to finish the course by presenting Vinogradov's three-primes theorem - this is more technical and complicated than the earlier topics, but it is also very interesting and should make for a satisfying conclusion.

There will be no final exam or midterms. Instead, each student taking this course for credit will be expected to present a short part of the course material (1-2 hours). The presentation topics can be chosen from the list below; if there is something else that you would like to present here, please discuss it with me in advance.

There is a lot of active research going on in this field. Accordingly, we will look at both classical results and some very recent work. While the focus will be firmly on number theory, there will be various bits of combinatorics and geometry mixed in. You may also expect guest presentations from a few faculty and visitors working on closely related topics.

**Roth's theorem:**a subset of**Z**of positive density must contain a 3-term arithmetic progression.

Sources:- W.T. Gowers:
*A new proof of Szemeredi's theorem*, Geom. Funct. Anal. 11 (2001), 465-588; also available from Tim Gowers's web page (linked below)

- Roth's theorem for finite fields (ask Ben Green)
- Roth-type theorem for triangles in the plane (ask Jozsef Solymosi)
- Behrend's construction of a large set with no arithmetic progressions

- W.T. Gowers:
**Freiman's theorem:**a set of integers with a small sumset is contained in a generalized arithmetic progression. In addition to Fourier analysis, the proof also uses a fair bit of combinatorics (Plunnecke-Ruzsa inequalities) and geometry of numbers (Minkowski's theorem).

Sources:- B.J. Green:
*Structure Theory of Set Addition*, ICMS Edinburgh Instructional Conference lecture notes, 2002; available from Ben Green's web page (linked below) - Nathanson's book, vol. II.

- B.J. Green:
**Weyl's inequality:**a classical method of estimating exponential sums for uniformly distributed sequences.

Sources:- Gowers-Verstraete lecture notes, see below.
- Montgomery's book.

**The Waring problem for squares:**A set A of nonnegative integers is called a basis of order k for**N**if every positive integer can be written as a sum of k elements of A. Lagrange's theorem states that every positive integer is a sum of four squares (i.e. squares are a basis of order 4). Waring's problem is to extend this to cubes and higher powers of integers. We will approach the problem via the "circle method", which also provides an estimate on the number of representations of integers as sums of squares.

Sources:- Nathanson's book, vol. I.
- Vaughan's book.

**Schnirelman-Goldbach theorem:**primes are a basis of finite order. The proof includes an application of the Selberg sieve.

Sources:- Nathanson's book, vol. I.

**Vinogradov's three-primes theorem:**every sufficiently large odd integer is a sum of three primes. The proof incorporates many of the techniques seen earlier in the course, and much more.

Sources:- Gowers-Verstraete lecture notes.
- Davenport's book.
- Nathanson's book, vol. I.

**Szemeredi's theorem for 4-term arithmetic progressions,**if time allows. It is interesting to see why this is so much more difficult than Roth's theorem, which we covered in the first 2 classes.

Sources:- Gowers-Verstraete lecture notes.

- Notes from Tim Gowers's 1999 Cambridge course in additive and combinatorial number theory, written by Tim Gowers and Jacques Verstraete, are available from Jacques Verstraete's web page. The course has substantial overlap with this one.
- Tim Gowers has many other interesting things on his web page, including expository papers on additive and combinatorial number theory.
- Ben Green has written expositions of various topics of interest to us, such as Freiman's theorem and sieve techniques.

- H. Davenport,
*Multiplicative Number Theory*, 2nd ed. (revised by H. Montgomery), Springer-Verlag 1980. - G.A. Freiman:
*Foundation of a structural theory of set addition*, AMS, 1973 - 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. - R.C. Vaughan,
*The Hardy-Littlewood Method*, 2nd ed., Cambridge Univ. Press, 1997 (note: the 2nd edition is substantially different from the 1st).