Locally integrable cross sections and their intersection covolume

Abstract

We study systematically cross sections of probability preserving actions of unimodular groups and their associated transverse measures, and introduce the invariant \emph{intersection covolume} to quantify their periodicity. Our main theorem, derived from a higher order version of Kac's lemma, shows that the intersection covolume is bounded below by the intensity, with equality precisely when the action is induced by a lattice (in the sense of Mackey). We further prove that the natural cross sections of cut--and--project actions have finite intersection covolume.

26 pages, 2 figures. Companion to: "Periodicity of Point Processes in Abelian Groups without Lattices" arXiv:2509.20847