A Roth theorem for amenable groups

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∈Φn φ(T_gx)ψ(S_gT_gx) exist in L¹(X, A, μ) for any φ, ψ, ∈ L²(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 = 2z 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)}.