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Lower bound in the Roth theorem for amenable groups

Published online by Cambridge University Press:  03 July 2014

QING CHU  and
PAVEL ZORIN-KRANICH Open the ORCID record for PAVEL ZORIN-KRANICH [Opens in a new window]
QING CHU
Affiliation:
Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, USA email [email protected]
PAVEL ZORIN-KRANICH
Affiliation:
Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, 91904, Israel email [email protected]
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 6 , September 2015 , pp. 1746 - 1766
Copyright
© Cambridge University Press, 2014 

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