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Bounded distance equivalence of cut-and-project sets and equidecomposability

Sigrid Grepstad

TL;DR

The paper proves that for a lattice Γ ⊂ ℝ^m × ℝ^n and Jordan measurable windows W,W' ⊂ ℝ^n of equal measure, Λ(Γ,W) ≈_BD Λ(Γ,W') implies W and W' are p_2(Γ)-equidecomposable up to measure zero, linking BD-equivalence to window geometry. The proof lifts model sets to lattice subsets Γ_W, Γ_W', constructs a finite set of translations in p_2(Γ) to partition W into pieces, and shows these pieces reassemble W' up to measure-zero discrepancy. The work connects bounded distance to bounded remainder set theory, and, using Hadwiger invariants, provides an explicit description of BD-classes in hulls of simple quasicrystals, with corollaries for unions of intervals and parallelotope windows in low dimensions. A corrigendum addresses a gap in the main proof, clarifying the scope and consequences of the main theorem. Overall, the study clarifies how window equidecomposability governs dynamical BD-classes in model-set hulls, with implications for the structure of aperiodic order.

Abstract

We show that given a lattice $Γ\subset \mathbb{R}^m \times \mathbb{R}^n$, and projections $p_1$ and $p_2$ onto $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively, cut-and-project sets obtained using Jordan measurable windows $W$ and $W'$ in $\mathbb{R}^n$ of equal measure are bounded distance equivalent only if $W$ and $W'$ are equidecomposable by translations in $p_2(Γ)$. As a consequence, we obtain an explicit description of the bounded distance equivalence classes in the hulls of simple quasicrystals. A corrigendum is appended at the end of the paper.

Bounded distance equivalence of cut-and-project sets and equidecomposability

TL;DR

The paper proves that for a lattice Γ ⊂ ℝ^m × ℝ^n and Jordan measurable windows W,W' ⊂ ℝ^n of equal measure, Λ(Γ,W) ≈_BD Λ(Γ,W') implies W and W' are p_2(Γ)-equidecomposable up to measure zero, linking BD-equivalence to window geometry. The proof lifts model sets to lattice subsets Γ_W, Γ_W', constructs a finite set of translations in p_2(Γ) to partition W into pieces, and shows these pieces reassemble W' up to measure-zero discrepancy. The work connects bounded distance to bounded remainder set theory, and, using Hadwiger invariants, provides an explicit description of BD-classes in hulls of simple quasicrystals, with corollaries for unions of intervals and parallelotope windows in low dimensions. A corrigendum addresses a gap in the main proof, clarifying the scope and consequences of the main theorem. Overall, the study clarifies how window equidecomposability governs dynamical BD-classes in model-set hulls, with implications for the structure of aperiodic order.

Abstract

We show that given a lattice , and projections and onto and respectively, cut-and-project sets obtained using Jordan measurable windows and in of equal measure are bounded distance equivalent only if and are equidecomposable by translations in . As a consequence, we obtain an explicit description of the bounded distance equivalence classes in the hulls of simple quasicrystals. A corrigendum is appended at the end of the paper.
Paper Structure (8 sections, 11 theorems, 60 equations, 1 figure)

This paper contains 8 sections, 11 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bounded distance equivalence in the hull of a cut-and-project set
  3. Preliminaries
  4. Cut-and-project sets
  5. Equidecomposability of polytopes and Hadwiger invariants
  6. Proof of Theorem \ref{['thm:main']}
  7. Connection to bounded remainder sets
  8. Explicit description of bounded distance equivalence classes

Key Result

Theorem 1.1

Let $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$ be a lattice and let $W$ and $W'$ be bounded, Jordan measurable sets in $\mathbb{R}^n$ of equal measure. If the model sets $\Lambda(\Gamma, W)$ and $\Lambda(\Gamma, W')$ are bounded distance equivalent, then the window sets $W$ and $W'$ are $p_2(

Figures (1)

  • Figure 1: The polygon $S$ considered in Example \ref{['ex:invar']}, as well as the line $l_1$ defining a rank-$1$ invariant, and the line and point $(l_2, p)$ defining a rank-$0$ invariant in $\mathbb{R}^2$.

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1.1
  • Theorem 1.2: FGS2022SS2021
  • Corollary 1.3
  • Corollary 1.4
  • Example 1
  • Theorem 1.5
  • Conjecture 1.6
  • Corollary 1.7
  • Definition 2
  • ...and 13 more