Periodicity of point processes in abelian groups without lattices

2025/09/25 by Nachi Avraham-Re'em, Avraham-Re'em, Nachi, Michael Björklund +3 · 1 citation
Materials Science · #11B13 #11B57 #22B05 #22D40 #22F10 #28D15 #37A15 (primary) #43A70 #52C23 #60G55 #Abelian group #Bounded function #Class (philosophy) #Dynamical Systems (math.DS) #Elementary abelian group #FOS: Mathematics #Farey sequence #Group (periodic table) #Intersection (aeronautics) #Probability (math.PR) #Quasicrystal Structures and Properties #Real line

paper · pdf · doi:10.48550/arxiv.2509.20847

openalex publication_date 2025/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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.

Citations

• Locally integrable cross sections and their intersection covolume
• Intersection spaces and multiple transverse recurrence
• Sets of transfer times with small densities
• Spectral theory of approximate lattices in nilpotent Lie groups
• Approximate invariance for ergodic actions of amenable groups
• An introduction to higher energies and sumsets

Cited by

• Locally integrable cross sections and their intersection covolume

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