Burago and Kleiner and, independently, McMullen, gave examples of Delone sets (that is, subsets of Euclidean space that are discrete and separated in a uniform way) that are non-bi-Lipschitz equivalent to the standard lattice. We refine their methods of construction via a discretization technique, thus giving the first examples of Delone sets as above that are also repetitive, in the sense that a translated copy of each patch appears in every large enough ball.
"Some examples of repetitive, nonrectifiable Delone sets." Geom. Topol. 20 (4) 1909 - 1939, 2016. https://doi.org/10.2140/gt.2016.20.1909