A Half–commutative Ip Roth Theorem

This result strengthens a previous theorem, again due to Furstenberg and Katznelson ([FK1]), stating that for commuting measure preserving transformations T1, . . . , Tk of a probability space (X,A, μ) and A ∈ A with μ(A) > 0 there exists n > 0 such that μ ( A∩ ⋂k i=1 T −n i A ) > 0. The case of [FK1] in which each Ti is a power of the same transformation T was proved by Furstenberg in 1977 ([F]), and is equivalent to Szemerédi’s theorem ([Sz]) on existence of arithmetic progressions in positive density subsets of N. (In particular, Theorem A provides that n may be chosen from any IP-system in N.) For this reason, Furstenberg and Katznelson refer to Theorem A as an “IP Szemerédi theorem”. The special case k = 2 of Theorem A might therefore be called an “IP Roth theorem”, for it may be used to infer the case of Szemerédi’s theorem dealing with three-term arithmetic progressions, which is due to K. Roth ([R]). Our goal here is to remove some of the commutativity restrictions on Theorem A in the special case k = 2 (hence the title of the paper). There are at least two reasonable choices for the definition of “IP-system” (Tα)α∈F in a non-abelian semigroup. We may require either Tα∪β = TαTβ or Tα∪β = TβTα for α, β ∈ F