  The purpose of this paper is to study the phenomenon of large intersections
in the framework of multiple recurrence for measure-preserving actions of
countable abelian groups. Among other things, we show: (1) If $G$ is a
countable abelian group and $\varphi, \psi : G \to G$ are homomorphisms such
that $\varphi(G)$, $\psi(G)$, and $(\psi - \varphi)(G)$ have finite index in
$G$, then for every ergodic measure-preserving system $(X, \mathcal{B}, \mu,
(T_g)_{g \in G})$, every set $A \in \mathcal{B}$, and every $\varepsilon > 0$,
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. (2) If $G$ is a countable abelian group
and $r,s \in \mathbb{Z}$ are integers such that $rG$, $sG$, and $(r \pm s)G$
have finite index in $G$, then for every ergodic measure-preserving system $(X,
\mathcal{B}, \mu, (T_g)_{g \in G})$, every set $A \in \mathcal{B}$, and every
$\varepsilon > 0$, the set $\{g \in G : \mu(A \cap T_{rg}^{-1}A \cap
T_{sg}^{-1}A \cap T_{(r+s)g}^{-1}A) > \mu(A)^4 - \varepsilon\}$ is syndetic. In
particular, these extend and generalize results of Bergelson, Host, and Kra
concerning $\mathbb{Z}$-actions and of Bergelson, Tao, and Ziegler concerning
$\mathbb{F}_p^{\infty}$-actions. Using an ergodic version of the Furstenberg
correspondence principle, we obtain new combinatorial applications. We also
discuss numerous examples shedding light on the necessity of the various
hypotheses above. Our results lead to a number of interesting questions and
conjectures, formulated in the introduction and at the end of the paper.
