A DICHOTOMY FOR BOUNDED DISPLACEMENT AND CHABAUTY-FELL CONVERGENCE OF DISCRETE SETS

The paper’s Theorem 1.1 proves the dichotomy for a compact minimal Delone dynamical system X: either X contains a uniformly spread set (so all are uniformly spread and BD(X)=1) or X contains continuously many BD-inequivalent Delone sets (BD(X)=2^{aleph0}) . The proof proceeds via a necessary-and-sufficient discrepancy criterion for BD-non-equivalence (Theorems 2.2–2.3) , a minimality-based uniform return lemma and a patch convergence theorem (Lemma 3.2 and Theorem 3.3) , and a combinatorial construction in §5 that yields a continuum of BD-classes (using (5.1)–(5.5) and concluding via (5.15)) . The candidate (model) solution proves the same dichotomy by a different route: it shows each bounded-BD subrelation E_n is closed and nowhere dense, hence E is meager, then applies Mycielski’s theorem to obtain a perfect set of pairwise BD-inequivalent Delone sets; it also observes that if X contains one uniformly spread set, then all are uniformly spread. This matches the paper’s conclusion, though the techniques differ (paper: quantitative discrepancy and patch limits; model: Baire-category and descriptive set theory).