A dichotomy for bounded displacement and Chabauty-Fell convergence of discrete sets

We prove that in every compact space of Delone sets in Rd which is minimal with respect to the action by translations, either all Delone sets are uniformly spread, or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty-Fell topology on the space of closed subsets of Rd. This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.
A main step in the proof is a result concerning Delone sets as limits of converging sequences of finite patches with respect to the Chabauty-Fell topology, under the assumption of minimality. In the infinite local complexity setting, information on a converging sequence does not immediately imply information regarding finite patches in the limit Delone set, and we provide sufficient conditions under which certain qualitative and
quantitative information can be deduced.