Fast repetitivity in non-rectifiable Delone sets

TL;DR

This paper constructs non-rectifiable, repetitive Delone sets in all Euclidean spaces and determines a near-optimal repetitivity function, advancing understanding of aperiodic order in geometric structures.

Contribution

It introduces a method to construct non-rectifiable, repetitive Delone sets in any Euclidean space and establishes a near-optimal repetitivity function for these sets.

Findings

Existence of non-rectifiable, repetitive Delone sets in all dimensions

Development of a near-optimal repetitivity function

Application of encoding non-realisable density in Delone sets

Abstract

We present a construction of non-rectifiable, repetitive Delone sets in every Euclidean space $\mathbb{R}^d$ with $d \geq 2$. 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.