Ergodic Methods in Additive Combinatorics

Shortly after Szemerédi’s proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of combinatorial ergodic theory, in which problems motivated by additive combinatorics are addressed kwith ergodic theory. Combinatorial ergodic theory has since produced combinatorial results, some of which have yet to be obtained by other means, and has also given a deeper understanding of the structure of measure preserving systems. We outline the ergodic theory background needed to understand these results, with an emphasis on recent developments in ergodic theory and the relation to recent developments in additive combinatorics. These notes are based on four lectures given during the School on Additive Combinatorics at the Centre de recherches mathématiques, Montreal in April, 2006. The talks were aimed at an audience without background in ergodic theory. No attempt is made to include complete proofs of all statements and often the reader is referred to the original sources. Many of the proofs included are classic, included as an indication of which ingredients play a role in the developments of the past ten years. 1. Combinatorics to ergodic theory 1.1. Szemerédi’s theorem. Answering a long standing conjecture of Erdős and Turán [11], Szemerédi [54] showed that a set E ⊂ Z with positive upper density contains arbitrarily long arithmetic progressions. Soon thereafter, Furstenberg [16] gave a new proof of Szemerédi’s Theorem using ergodic theory, and this has lead to the rich field of combinatorial ergodic theory. Before describing some of the results in this subject, we motivate the use of ergodic theory for studying combinatorial problems. We start with the finite formulation of Szemerédi’s theorem: Theorem 1.1 (Szemerédi [54]). Given δ > 0 and k ∈ N, there is a function N(δ, k) such that if N > N(δ, k) and E ⊂ {1, . . . , N} is a subset with |E| ≥ δN , then E contains an arithmetic progression of length k. 1991 Mathematics Subject Classification. Primary 37A30; Secondary 11B25, 27A45.