Extension of Khintchine's recurrence in a simple case Suppose an ergodic system $(X,\mathcal{B},\mu,T)$ has a Kronecker factor that is isomorphic to an ergodic rotation, say on the Torus.
How can one prove that the large intersection property holds for $T$, namely that for $A \in \mathcal{B}$ with positive measure and $ε>0$ there exists $n \in \mathbb{N}$ (in fact a syndetic set) s.t. $\mu(A\cap Τ^{-n}A\cap T^{-2n}A)\geq (\mu(A))^3-ε $?
If the Kronecker factor is $Zx=x+α$, with $α \in \mathbb{R}-\mathbb{Q}$ then the above property can easily be seen to hold for $n$ in a syndetic subset of $\mathbb{N}$, by the equidistribution of $(\{nα\})_{n \in \mathbb{N}}$. But how can we proceed for the initial system? I assume we could use some sort of a weighted average and show that the Kronecker factor is-in a sense-characteristic for these averages.
 A: (Edit: my original remark didn't take all of the question into account.  One can say that the Kronecker factor does retain sufficient detail to prove the desired result.)
The large intersection property you ask for was first established by Bergelson, Host, and Kra [1].  Frantzikinakis [2] gave a simplified proof and extended the result to polynomial configurations.  Ackelsberg, Bergelson, and Best [3]
have extended the result to actions of countable abelian groups, and the introduction there has a nice overview of the history and technical issues involved.  I believe [2] and [3] use the method you describe.
[1] Bergelson, Vitaly; Host, Bernard; Kra, Bryna, Multiple recurrence and nilsequences (with an appendix by Imre Ruzsa), Invent. Math. 160, No. 2, 261-303 (2005). ZBL1087.28007.
[2] Frantzikinakis, Nikos, Multiple ergodic averages for three polynomials and applications, Trans. Am. Math. Soc. 360, No. 10, 5435-5475 (2008). ZBL1158.37006.
[3] Ackelsberg, Ethan; Bergelson, Vitaly; Best, Andrew, Multiple recurrence and large intersections for abelian group actions,  ZBL07471818.
