Delone sets that are not rectifiable under Lipschitz co-uniformly continuous bijections

TL;DR

This paper demonstrates the existence of Delone sets in higher-dimensional Euclidean spaces that cannot be transformed into standard lattices via Lipschitz co-uniformly continuous bijections with controlled asymptotic distortion.

Contribution

It introduces the first examples of Delone sets that are non-rectifiable under specific Lipschitz co-uniformly continuous maps with asymptotic distortion control.

Findings

Existence of non-rectifiable Delone sets in $\\mathbb{R}^d$ for $d \geq 2$.

Use of Lipschitz regular maps concepts by Dymond, Kaluža, and Kopecká.

Impossibility results for mapping certain Delone sets onto lattices under specified conditions.

Abstract

We prove that there exist Delone sets in $\mathbb{R}^d$, $d \geq 2$, which cannot be mapped onto the standard lattice $\mathbb{Z}^d$ by Lipschitz co-uniformly continuous bijections satisfying an asymptotic control on the lower distortion. The impossibility of the unrectifiability crucially uses ideas of Lipschitz regular maps recently introduced by M. Dymond, V. Kalu\v{z}a and E. Kopeck\'a.