  The goal of this paper is to generalize, refine, and improve results on large
intersections. 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, for
any 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 $$\{g\in G : \mu(A\cap
T_{\varphi(g)}^{-1} A\cap T_{\psi(g)}^{-1}A)>\mu(A)^3-\varepsilon\}$$ is
syndetic. 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 fails
in general if all three groups are of infinite index. One particularly
interesting case is where $G=(\mathbb{Q}_{>0},\cdot)$ and $\varphi(g)=g$,
$\psi(g)=g^2$, which leads to a multiplicative version for the large
intersection result of Bergelson-Host-Kra. We also completely characterize the
pairs of homomorphisms $\varphi,\psi$ that have the large intersections
property when $G=\mathbb{Z}^2$. The proofs of our main results rely on analysis
of the structure of the \emph{universal characteristic factor} for the multiple
ergodic averages $$\frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot
T_{\psi(g)} f_2.$$ In the case where $G$ is finitely-generated, the
characteristic factor for such averages is the \emph{Kronecker factor}. In this
paper, we study actions of groups that are not necessarily finitely-generated,
showing in particular that by passing to an extension of $X$, one can describe
the characteristic factor in terms of the \emph{Conze--Lesigne factor} and the
$\sigma$-algebras of $\varphi(G)$ and $\psi(G)$ invariant functions.
