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:

Tuesday, October 30:

Wednesday, October 31: