Density functions of periodic sequences

TL;DR

This paper provides a complete theoretical description of the density functions of periodic sequences, confirming their properties and symmetries, and clarifying previous computational observations.

Contribution

It offers a full characterization of density functions for any periodic sequence, advancing understanding of their invariance and symmetry properties.

Findings

Explicit description of density functions for all periodic sequences

Confirmation of symmetry properties of these functions

Theoretical validation of previously observed coincidences

Abstract

Periodic point sets model all solid crystalline materials whose structures are determined in a rigid form and should be studied up to rigid motion or isometry preserving inter-point distances. In 2021 H.Edelsbrunner et al. introduced an infinite sequence of density functions that are continuous isometry invariants of periodic point sets. These density functions turned out to be highly non-trivial even in dimension 1 for periodic sequences of points in the line. This paper fully describes the density functions of any periodic sequence and their symmetry properties. The explicit description theoretically confirms coincidences of density functions that were previously computed only through finite samples.