Non-rectifiable Delone sets under pointwise co-Lipschitz bijections

Abstract

For each $d\in {\mathbb{N}}_{\ge 2}$ we construct a Delone set $Y$ in ${\mathbb{R}}^d$ for which every Lipschitz bijection from $Y$ to ${\mathbb{Z}}^d$ has a very irregular inverse. For example, the inverse fails to be Lipschitz, even at just a single point. Further, we clarify the relationship between several notions of regularity for bijections between Delone sets, studied in the literature.