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Lower bound in the Roth theorem for amenable groups
Published online by Cambridge University Press: 03 July 2014
Abstract
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.
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- © Cambridge University Press, 2014
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