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The goal of this paper is to generalize, refine, and improve results on largeintersections. We show that if $G$ is a countable abelian group and $\varphi,\psi : G \to G$ are homomorphisms such that at least two of the three subgroups$\varphi(G)$, $\psi(G)$, and $(\psi-\varphi)(G)$ have finite index in $G$, then$\{\varphi, \psi\}$ has the \emph{large intersections property}. That is, forany ergodic measure preserving system $X=(X,\mathcal{X},\mu,(T_g)_{g\in G})$,any $A\in\mathcal{X}$, and any $\varepsilon>0$, the set $$\{g\in G : \mu(A\capT_{\varphi(g)}^{-1} A\cap T_{\psi(g)}^{-1}A)>\mu(A)^3-\varepsilon\}$$ issyndetic. Moreover, in the special case where $\varphi(g)=ag$ and $\psi(g)=bg$for $a,b\in\mathbb{Z}$, we show that we only need one of the groups $aG$, $bG$,or $(b-a)G$ to be of finite index in $G$, and we show that the property failsin general if all three groups are of infinite index. One particularlyinteresting case is where $G=(\mathbb{Q}_{>0},\cdot)$ and $\varphi(g)=g$,$\psi(g)=g^2$, which leads to a multiplicative version for the largeintersection result of Bergelson-Host-Kra. We also completely characterize thepairs of homomorphisms $\varphi,\psi$ that have the large intersectionsproperty when $G=\mathbb{Z}^2$. The proofs of our main results rely on analysisof the structure of the \emph{universal characteristic factor} for the multipleergodic averages $$\frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdotT_{\psi(g)} f_2.$$ In the case where $G$ is finitely-generated, thecharacteristic factor for such averages is the \emph{Kronecker factor}. In thispaper, we study actions of groups that are not necessarily finitely-generated,showing in particular that by passing to an extension of $X$, one can describethe characteristic factor in terms of the \emph{Conze--Lesigne factor} and the$\sigma$-algebras of $\varphi(G)$ and $\psi(G)$ invariant functions.

Khintchine-type recurrence for 3-point configurations