Khintchine-type double recurrence in abelian groups

We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if $\Gamma$ is a countable discrete abelian group, $\varphi, \psi \in End(\Gamma)$, and $\psi - \varphi$ is an injective endomorphism with finite index image, then for any ergodic measure-preserving $\Gamma$-system $\left( X, \mathcal{X}, \mu, (T_g)_{g \in \Gamma} \right)$, any measurable set $A \in \mathcal{X}$, and any $\varepsilon > 0$, the set of $g \in \Gamma$ for which $$\mu \left( A \cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A \right) > \mu(A)^3 - \varepsilon$$ is syndetic. This generalizes the main results of (Ackelsberg--Bergelson--Shalom, 2022) and essentially answers a question left open in that paper (Question 1.12). For the group $\Gamma = \mathbb{Z}^d$, we deduce that for any matrices $M_1, M_2 \in M_{d \times d}(\mathbb{Z})$ whose difference $M_2 - M_1$ is nonsingular, any ergodic measure-preserving $\mathbb{Z}^d$-system $\left( X, \mathcal{X}, \mu, (T_{\vec{n}})_{\vec{n} \in \mathbb{Z}^d} \right)$, any measurable set $A \in \mathcal{X}$, and any $\varepsilon > 0$, the set of $\vec{n} \in \mathbb{Z}^d$ for which $$\mu \left( A \cap T_{M_1 \vec{n}}^{-1} A \cap T_{M_2 \vec{n}}^{-1} A \right) > \mu(A)^3 - \varepsilon$$ is syndetic, a result that was previously known only in the case $d = 2$. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom that says that the relevant ergodic averages are controlled by a characteristic factor closely related to the quasi-affine (or Conze--Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to $\varphi$ and $\psi$) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum.