Automatic sets and Delone sets

Automatic sets D subset of Z(m) are characterized by having a finite number of decimations. They are equivalently generated by fixed points of certain substitution systems, or by certain finite automata. As examples, two-dimensional versions of the Thue-Morse, Baum-Sweet, Rudin-Shapiro and paperfolding sequences are presented. We give a necessary and sufficient condition for an automatic set D subset of Z(m) to be a Delone set in R-m. The result is then extended to automatic sets that are defined as fixed points of certain substitutions. The morphology of automatic sets is discussed by means of examples.