Fast repetitivity in non-rectifiable Delone sets

We present a construction of non-rectifiable, repetitive Delone sets in every Euclidean space ${\mathbb{R}}^d$ with $d\ge 2$. We further obtain a close to optimal repetitivity function for such sets. The proof is based on the process of encoding a non-realisable density in a Delone set, due to Burago and Kleiner.