This semester, we meet on Wednesdays,

The plan is to 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. These are all hot research topics - accordingly, we may often want to progress as soon as possible to reading current research papers, some not yet published in journals. Research problems, of varying levels of difficulty, will be suggested and discussed.

- Fourier analysis: "randomness" criteria for sets, finding structure in sets that are not random.
- combinatorial approach: Szemeredi's regularity lemma, its extensions and applications,
- ergodic-theoretic approach: background, multiple recurrence theorem, generalizations of Szemeredi's theorem,
- quantitative results for 3-term arithmetic progressions,
- the recent work of Green and Tao on arithmetic progressions in primes,
- other related results, e.g. finding patterns in subsets of the integer lattice.

- The Kakeya problem: the number-theoretic aspects. (The Fourier-analytic aspects require a fair bit of background in harmonic analysis, and hopefully will be included in a separate topics course some time in the future.)
- Sums vs. products: if A is a set of n numbers, must at least one of
the sets {a+b: a,b in A} and {ab: a,b in A} have cardinality
(almost) n
^{2}? - Distance set problems (example: what is the minimum number of distinct distances between n points in the plane?).
- And many others of similar flavour: easy to state, difficult to solve, often require combining seemingly unrelated methods and ideas.

The entire course (2 semesters) is worth 4 credits if you sign up. However, you do not have to sign up in order to attend. (Since this is a reading course, there is no minimum number of students required.)

- Research papers, expository papers and lecture notes can be downloaded from the web pages of Tim Gowers, Ben Green, and Terry Tao.
- Number theory books:
- 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. - 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,
- Papers and lecture notes on the Web:
- "Recent Trends in Additive Combinatorics", AIM workshop, September 2004: web page includes open problems contributed by participants before the workshop. Currently in preparation are notes from the open problems sessions.
- A new proof of Szemeredi's theorem, by Tim Gowers.
- The primes contain arbitrarily long arithmetic progressions, by Ben Green and Terry Tao.
- Structure theory of set addition, by Ben Green (an exposition of Freiman's theorem)
- (Gowers)-Balog-Szemeredi theorem, by Vsevolod Lev (an exposition)
- Additive combinatorics lecture notes, by Terry Tao (includes exposition of Szemer\'edi's theorem for progressions of length 4)

- More papers, to be discussed in the second part of the course:
- On one problem of Gowers, by I.D. Shkredov: analytic proof of Roth's theorem for triangles in subsets of a lattice.
- Finite field models in additive combinatorics, by B. Green: finite field versions of various number-theoretic problems, e.g. Roth's theorem.
- An argument of Shkredov in the finite field setting, by B. Green: an addendum to the previous paper, concerning "Roth for triangles".
- Lectures in Harmonic Analysis, by T. Wolff: download it here, or buy the book from AMS. Chapters 10 and 11 include an introduction to the Kakeya problem, in particular the Kakeya finite fields problem is introduced in Chapter 11.
- A new bound on partial sum-sets and difference sets, and applications to the Kakeya conjecture, by N. Katz and T. Tao: a possible source for the "arithmetic" approach to Kakeya, though the main ideas are perhaps better explained in the next paper.
- New bounds on Kakeya problems, by N. Katz and T. Tao: the "arithmetic approach" developed further.
- Restriction and Kakeya phenomena for finite fields, by G. Mockenhaupt and T. Tao: another good starting point for the Kakeya finite fields problem.
- A new bound for finite field Besicovitch sets in four dimensions, by T. Tao: more finite-field Kakeya.
- A sum-product estimate in finite fields and applications, by J. Bourgain, N. Katz, T. Tao.