  Lattices and periodic point sets are well known objects from discrete
geometry. They are also used in crystallography as one of the models of atomic
structure of periodic crystals. In this paper we study the embedding properties
of spaces of lattices and periodic point sets equipped with optimal bijection
metrics (i.e., bottleneck and Euclidean bottleneck metrics). We focus our
treatment on embeddings into Hilbert space. On one hand this is motivated by
modern data analysis, which is mostly based on statistical approaches developed
on Euclidean on Hilbert spaces, hence such embeddings play a major role in
applied pipelines. On the other hand there is a well-established methodology
related to such questions in coarse geometry, arising from the work on the
Novikov conjecture. The main results of this paper provide different
conditions, under which the spaces of lattices or periodic point sets are
Lipschitz or coarsely (non)embeddable into Hilbert space. The various
conditions are phrased in terms of density, packing radius, covering radius,
the cardinality of the motif, and the diameter of the unit cell.
