Periodicity of point processes in abelian groups without lattices

We investigate the intersection covolume of cross sections for probability preserving actions of a class of abelian groups without lattices, including $p$-adic groups and the group of finite adeles. We show that for cross sections with a uniformly discrete return time set, the intersection covolume is bounded below by twice the intensity, revealing a strict gap compared to the lattice case. Our main theorem asserts that cut-and-project systems uniquely attain the minimal intersection covolume. As an application, we characterize the generalized Farey fractions in the finite adeles via their Banach density.