A Roth theorem for amenable groups

Abstract

We prove the following mean ergodic theorem: for any two commuting measure preserving actions { T g } and { S g } of a countable amenable group G on a probability space ( X, A , μ), lim n →∞ 1/|Φ n | Σ g ∈Φ n φ( T g x )ψ( S g T g x ) exist in L 1 ( X, A , μ) for any φ, ψ, ∈ L 2 ( X, A , μ), where {Φ n } is any left Følner sequence for G . This generalizes Furstenberg's ergodic Roth theorem, which corresponds to the case G = Z , T g = S g , as well as a more general result of Conze and Lesigne (which corresponds to the case G = Z with no restrictions on T g and S g ). The limit is identified, and two combinatorial corollaries are obtained. The first of these states that in any subset E ⊂ G × G which is of positive upper density (with regard to any left Følner sequence in G × G ), we may find triangular configurations of the form {( a, b ), ( ga, b ), ( ga, gb )}. This result has as corollaries Roth's theorem on arithmetic progressions of length three and a theorem of Brown and Buhler guaranteeing solutions to the equation x + y = 2 z in any sufficiently big subset of an abelian group of odd order. The second corollary states that if G × G × G is partitioned into finitely many cells, one of these cells contains configurations of the form {( a, b, c ), ( ga, b, c ,), ( ga, gb, c ), ( ga, gb, gc )}.