The asymptotic behaviour and a near linear time algorithm for isometry invariants of periodic sets

Abstract: The fundamental model of a periodic structure is a periodic point set up to rigid motion or isometry. Our paper in SoCG 2021 defined isometry invariants (density functions), which are complete in general position and continuous under perturbations. This work introduces much faster isometry invariants (average minimum distances), which are also continuous and distinguish some sets that have identical density functions. We explicitly describe the asymptotic behaviour of the new invariants for a wide class of sets including non-periodic. The proposed near linear time algorithm processed a dataset of hundreds of thousands of real structures in a few hours on a modest desktop.