The ergodic and combinatorial approaches to Szemerédi's theorem

Details The ergodic and combinatorial approaches to Szemerédi's theorem The ergodic and combinatorial approaches to Szemerédi's theorem

2006-04-20

arxiv.org/abs/math/0604456v1

Mathematics

Combinatorics

48 pages, no figures. Based on a lecture series given at the Montreal Workshop on Additive Combinatorics, April 6-12 2006

Scientific paper

A famous theorem of Szemer\'edi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szemer\'edi, the ergodic theory approach of Furstenberg, the Fourier-analytic approach of Gowers, and the hypergraph approach of Nagle-R\"odl-Schacht-Skokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different.

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