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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.

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(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.

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  • $\begingroup$ Thank you for the answer. Indeed, the average you mention might be smaller than $\mu(A)^3$, but one can instead use a weighted average for the Kronecker factor which is larger than the same bound. So the question becomes; Is the Kronecker factor characteristic for the weighted averages as well? $\endgroup$
    – User
    Commented Mar 30, 2022 at 7:22
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    $\begingroup$ I should have read more carefully. I believe the method you describe was used in Frantzikinakis's article referenced above, and in Section 8 of the Ackelsberg, Bergelson, Best article. $\endgroup$
    – John Griesmer
    Commented Mar 30, 2022 at 13:18
  • $\begingroup$ Yes, I believe this is the context of lemma 8.1 in [3] (in a more general setting), but I am not familiar with the notation and some definitions used, so I can't really understand the proof at the moment. $\endgroup$
    – User
    Commented Mar 31, 2022 at 7:38

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