Recognition of near-duplicate periodic patterns by continuous metrics

 We develop the local theory initiated by Delone to recognize periodic  patterns under rigid motion in Euclidean geometry. The 3-dimensional  case is practically important for justifying the novelty of solid  crystalline materials (periodic crystals) and for patenting medical  drugs in a solid tablet form. Past descriptors based on finite subsets  fail when a unit cell of a periodic pattern discontinuously changes  under almost any perturbation of atoms, which is inevitable due to  noise and atomic vibrations. The major problem is not only to find  complete invariants but to design efficient algorithms for distance  metrics on these invariants that should continuously behave under  noise. The proposed continuous metrics solve this problem in any  Euclidean space and are algorithmically approximated with small error  factors in times that are explicitly bounded in the size and  complexity of a given pattern. The proved Lipschitz continuity allows  us to confirm all near-duplicates in major databases of experimental  and simulated crystals. This practical detection of noisy duplicates  should stop the artificial generation of ‘new’ materials from slight  perturbations of known crystals. Several such duplicates are being  investigated by five journals for data integrity. The talk is based on  the paper in arxiv:2205.15298, the latest version is available at  https://kurlin.org/projects/periodic-geometry/near-duplicate-periodic-patterns.pdf