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, $\varphi , \psi \in \mathrm {End}(\Gamma )$ , and $\psi - \varphi $ is an injective endomorphism with finite index image, then for any ergodic measure-preserving $\Gamma $ -system $( X, {\mathcal {X}}, \mu , (T_g)_{g \in \Gamma } )$ , any measurable set $A \in {\mathcal {X}}$ , and any ${\varepsilon }> 0$ , there is a syndetic set of $g \in \Gamma$ such that $\mu ( A \cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A ) > \mu(A)^3 - \varepsilon$ . This generalizes the main results of Ackelsberg et al [Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma10 (2022), Paper no. e107] and essentially answers a question left open in that paper [Question 1.12; Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma10 (2022), Paper no. e107]. For the group $\Gamma = {\mathbb {Z}}^d$ , the result applies to pairs of endomorphisms given by matrices whose difference is non-singular. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom [Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma10 (2022), Paper no. e107] 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.