Department Colloquium
Department of Mathematics
Princeton University
November 8th, 2000
Arithmetic progressions of length four
The famous theorem of Szemer\'edi, proving a conjecture of
Erd\"os and Tur\'an from 1936, asserts that for every
$\delta>0$ and every natural number $k$ there exists an
$N$ such that every subset $A\subset\{1,2,\dots,N\}$ of
cardinality at least $\delta N$ contains an arithmetic
progression of length $k$. When $k=2$ this assertion is
trivial. The case $k=3$ was proved by Roth in 1953 using
the circle method.
The case $k=4$ is much harder (for reasons that can be
made quite explicit) and was not solved until 1969, when
Szemer\'edi found a highly ingenious combinatorial argument,
which over the next few years he was able to extend to
progressions of arbitrary length. In 1977, Furstenberg
discovered a completely different and more conceptual
argument using ergodic theory, which led to many extensions
of the original theorem.
Both these proofs ignored Roth's method, and indeed there are
very serious obstacles to extending this method to progressions
of length greater than three, as I shall demonstrate. However,
it can be done, and this will be the main topic of the talk.
One of the main advantages of the new approach to the theorem
is that it gives very greatly improved bounds for the dependence
of $N$ on $k$ and $\delta$. Another is that the proof is much
more closely related to results in additive number theory and
may eventually lead to the solution of problems in that area.