Lower bound in the Roth theorem for amenable groups

Abstract

Let $T_1$, $T_2$ 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 $\mu (A\cap T_1^{g}A\cap T_1^g{T}_2^{g}A)>\mu (A{)}^4-\epsilon$ on a syndetic set for any measurable set $A$ and any $\epsilon >0$. The proof uses the concept of a sated system introduced by Austin.

v2: added a counterexample showing that the exponent in the main result cannot be improved to 3. To appear in Ergodic Theory and Dynamical Systems

References in corpus (2)

• Multiple recurrence and the structure of probability-preserving systems
• Multiparameter ergodic averages for two commuting actions of an amenable group

Cited by in corpus (3)

• Finite Products Sets and Minimally Almost Periodic Groups
• Triangles in cartesian squares of quasirandom groups
• Uniform syndeticity in multiple recurrence