"A dichotomy for bounded displacement equivalence"
Solomon, YaarDelone sets $Y$ and $Z$ are BD-equivalent if there exists a bounded displacement (BD) mapping between them, namely a bijection $f \colon Y \to Z$ such that the quantity $\|f(y)-y\|$ ($y \in Y$) is bounded. We study the cardinality of the set of equivalence classes and show that a minimal space of Delone sets either contains a set which is BD to a lattice, in which cases all points in the space are such, or there are continuously many BD-classes represented in the space. If time permits, we will discuss some applications of this dichotomy to several known constructions from the theory of aperiodic order. Based on a joint work with Yotam Smilansky.