Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fast repetitivity in non-rectifiable Delone sets

Ashwin Bhat, Michael Dymond

TL;DR

The paper proves that in every dimension $d\ge2$ there exist repetitive, non-rectifiable Delone sets in $\mathbb{R}^d$ that encode a non-realisable density $\rho$, with an explicit near-optimal repetitivity bound $R(r)\in\bigcap_{q>2/d} o\left(r\left(\frac{\log r}{\log\log r}\right)^q\right)$. The construction hinges on encoding $\rho$ via a hierarchical palette framework and translating this into a Delone set using a base repetitive map on $\mathbb{Z}^d$, building on Burago–Kleiner’s non-realisable-density technique. A key result shows that if an X-encoding $\rho$ is sufficiently repetitive in a certain sense, then $\rho$ must be constant almost everywhere, highlighting a fundamental limitation of the encoding method for achieving very fast repetitivity. Collectively, the work provides the first explicit repetitive, non-rectifiable Delone sets with concrete, near-optimal repetitivity bounds and clarifies the tension between density-encoding and rectifiability in higher dimensions.

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.

Fast repetitivity in non-rectifiable Delone sets

TL;DR

The paper proves that in every dimension there exist repetitive, non-rectifiable Delone sets in that encode a non-realisable density , with an explicit near-optimal repetitivity bound . The construction hinges on encoding via a hierarchical palette framework and translating this into a Delone set using a base repetitive map on , building on Burago–Kleiner’s non-realisable-density technique. A key result shows that if an X-encoding is sufficiently repetitive in a certain sense, then must be constant almost everywhere, highlighting a fundamental limitation of the encoding method for achieving very fast repetitivity. Collectively, the work provides the first explicit repetitive, non-rectifiable Delone sets with concrete, near-optimal repetitivity bounds and clarifies the tension between density-encoding and rectifiability in higher dimensions.

Abstract

We present a construction of non-rectifiable, repetitive Delone sets in every Euclidean space with . 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.

Paper Structure

This paper contains 11 sections, 14 theorems, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Sets and measures
  4. Delone sets
  5. Mappings
  6. A general construction of a repetitive Delone set
  7. A quick aside about colours and palettes
  8. Encoding a non-realisable density in a sequence of palettes
  9. Non-rectifiability and repetitivity
  10. Optimality of the repetitivity function
  11. Acknowledgements

Key Result

Theorem 1.1

[Burago and Kleiner burago1998separated, see also dymond2023highly] Let $X \subset \mathbb{R}^d$ be a Delone set and $\rho:[0,1]^d \to \mathbb{R}_{> 0}$ be a measurable function which is non-realisable. If $X$ encodes $\rho,$ then $X$ is non-rectifiable.

Figures (2)

  • Figure 1: Example of a palette of colours
  • Figure 2: How the palettes for $d=2$ are built at levels $n-1,$$n$ and $n+1$ in Theorem \ref{['intermediate thm']} depicted with step size $h_n = 1$. If a small square is coloured pink, then its bottom-left corner is assigned the value $1$, and if it is coloured red, its bottom-left corner is assigned the value $2$. The palette at level $n$ contains colours with every $4-$combination of colours from the $(n-1)^{th}$ palette along its base. Moreover, the colours at level $n$ consist increasing shades from $\phi_1^{(n)}$ to $\phi_{c_{n}-1}^{(n)},$ while $\phi_{c_n}^{(n)}$ is an approximation of the density $\rho.$ If we increase $h_n,$ then the palette at level $n$ becomes 'coarser'.

Theorems & Definitions (47)

  • Definition 1.0
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['1']}
  • Theorem 1.4
  • Definition 1.5: lagarias2003repetitive, Definition $1.1$
  • Definition 1.6: lagarias2003repetitive, $(1.2)$, Definition $1.5$
  • Definition 1.7
  • Lemma 3
  • ...and 37 more