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

Abstract

We prove that there exist Delone sets in ${\mathbb{R}}^d$, $d\ge 2$, 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. Kaluža and E. Kopecká.

arXiv comment15 pages, 5 Figures. Typos and somes errors were fixed. We changed the "document-clasess" in the file. We mainly added a lower distortion in the maps that we consider. Comments are welcome!! V3: final version. Fixed typos. Accepted for publications in JMAA