Number of Bounded Distance Equivalence Classes in Hulls of Repetitive Delone Sets

The paper proves the dichotomy by constructing “deviant” and “normal” patches using Laczkovich-type discrepancy bounds and then embedding these in the hull to obtain 2^{aleph_0} many pairwise non-bde Delone sets; see Theorem 1.1 and the concluding cardinality argument in the paper’s Section 3 . The candidate solution proves the same dichotomy by a descriptive set theory route: it shows the bounded-distance-equivalence relation is F_sigma (hence Borel), uses Kuratowski–Ulam to deduce meagerness off the trivial case, and then applies Mycielski’s theorem to produce a perfect antichain. This yields the same 1-or-continuum conclusion. The approaches are different and compatible; the model’s proof is concise but needs a bit more detail for the closedness of E_M (existence of limiting M-bijections), which is standard to supply under FLC via a finite-branching compactness/diagonal argument. Overall: both correct, different proofs.