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, phi, psi is an element of End (Gamma), and psi-phi is an injective endomorphism with finite index image, then for any ergodic measure-preserving Gamma-system (X, chi, mu,(T-g)(g is an element of Gamma)), any measurable set A is an element of chi, and any epsilon > 0, there is a syndetic set of g is an element of Gamma such that mu(A boolean AND T-phi(g)(-1) A boolean AND T-psi(g)(-1) A) > mu(A)(3) - epsilon. This generalizes the main results of Ackelsberg et al [Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma 10 (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. Sigma 10 (2022), Paper no. e107]. For the group Gamma= 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. Sigma 10 (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 phi and psi) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum.