Published online by Cambridge University Press: 03 July 2014
Let $T_{1}$ and
$T_{2}$ be two commuting probability measure-preserving actions of a countable amenable group such that the group spanned by these actions acts ergodically. We show that
${\it\mu}(A\cap T_{1}^{g}A\cap T_{1}^{g}T_{2}^{g}A)>{\it\mu}(A)^{4}-{\it\epsilon}$ on a syndetic set for any measurable set
$A$ and any
${\it\epsilon}>0$. The proof uses the concept of a sated system, introduced by Austin.