A Roth theorem for amenable groups


We prove the following mean ergodic theorem: for any two commuting measure preserving actions {Tg} and {Sg} of a countable amenable group G on a probability space (X, A, μ), limn→∞ 1/|Φn| ∑g∈Φn φ(Tgx)ψ(SgTgx) exists in L1(X, A, μ) for any φ, ψ ∈ L2(X, A, μ), where {Φn} is any left Følner sequence for G. This generalizes Furstenberg's ergodic Roth theorem, which corresponds to the case G = Z, Tg = Sg, as well as a more general result of Conze and Lesigne (which corresponds to the case G = Z with no restrictions on Tg and Sg). The limit is identified, and two combinatorial corollaries are obtained. The first of these states that in any subset E ⊂ G x G which is of positive upper density (with regard to any left Følner sequence in G x 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 x G x 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)}.

Publication Title

American Journal of Mathematics

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