Alexander Ziwet

University of Michigan
Department of Mathematics

2001 Ziwet Lectures

October 29-31, 4:10-5:00, 1360 East Hall





Szemeredi's theorem and related results

Timothy Gowers

Cambridge University and Princeton




Abstract

Szemer\'edi's theorem states that every set of integers of positive upper density contains arbitrarily long arithmetic progressions. Equivalently, for any $\delta > 0$ and any $k$, every subset of $\{1,2,...,N\}$ of size at least $\delta N$ contains an arithmetic progression of length $k$, provided only that $N$ is sufficiently large. Until recently, the only known proofs of the theorem (Szemer\'edi's and a different approach by Furstenberg using ergodic theory) gave almost no information about the dependence of $N$ on $k$ and $\delta$. The purpose of these three lectures will be to explain a new, more quantitative approach. I hope to organize the three lectures roughly as follows.

Lecture 1: I shall show how to prove the result for $k=3$ using exponential sums. This proof was discovered by Roth in 1953. I shall also present a simple example which seems to rule out any extension of Roth's methods to longer progressions.

Lecture 2: Despite the example presented in the first lecture, this one will contain an outline of a proof for $k=4$, building on Roth's method - but also introducing some new ingredients, most notably a theorem of Freiman about the structure of sets with small sumset. The proof I give can be extended to arbitrary $k$. Though the extension is, in places, not easy, the most interesting ideas of the proof are contained in the $k=4$ case.

Lecture 3: This will be a little more miscellaneous. I shall give some idea of a beautiful proof by Imre Ruzsa of Freiman's theorem. I also intend to discuss a few results and open problems in a similar spirit to Szemer\'edi's theorem.

Dependence: Obviously, the second lecture will to some extent depend on the first, particularly for its motivation. Apart from this, I will try to keep dependences between the lectures to a minimum. I will also do my best to make the lectures accessible to those with little prior knowledge of additive and combinatorial number theory.


Monday, October 29:

Lecture 1.
Reception following the lecture, East Hall Atrium.

Tuesday, October 30:

Lecture 2.

Wednesday, October 31:

Lecture 3.

The Ziwet lectures were established at the bequest of Alexander Ziwet, a professor in the U-M Mathematics Department during 1888-1925.