Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many. A very similar result is proven in arXiv:2011.00106 [math.MG].
Frettlöh, D., Garber, A., & Sadun, L. (2021). Number of bounded distance equivalence classes in hulls of repetitive Delone sets. ArXiv:2101.02514 [Math]. http://arxiv.org/abs/2101.02514
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