A Characteristic Factor for the 3-Term IP Roth Theorem in $\mathbb{Z}_3^\mathbb{N}$

Abstract

Let $\Omega = \bigoplus_{i=1}^\infty \mathbb{Z}_3$ and $e_i = (0, \dots, 0 , 1, 0, \dots)$ where the $1$ occurs in the $i$-th coordinate. Let $\mathscr{F}=\{ \alpha \subset \mathbb{N} : \varnothing \neq \alpha, \alpha \text{ is finite} \}$. There is a natural inclusion of $\mathscr{F}$ into $\Omega$ where $\alpha \in \mathscr{F}$ is mapped to $e_\alpha = \sum_{i \in \alpha} e_i$. We give a new proof that if $E \subset \Omega$ with $d^*(E) >0$ then there exist $\omega \in \Omega$ and $\alpha \in \mathscr{F}$ such that \[ \{ \omega, \omega+ e_\alpha, \omega + 2 e_\alpha \} \subset E.\]Our proof establishes that for the ergodic reformulation of the problem there is a characteristic factor that is a one step compact extension of the Kronecker factor.