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

TL;DR

This paper characterizes when cut-and-project sets are bounded distance equivalent, showing it depends on the equidecomposability of the windows used, and describes the classes of such equivalence in quasicrystal hulls.

Contribution

It establishes a precise criterion linking bounded distance equivalence of cut-and-project sets to the equidecomposability of their defining windows.

Findings

Bounded distance equivalence depends on window equidecomposability.

Explicit description of equivalence classes in quasicrystal hulls.

Provides conditions for when different cut-and-project sets are bounded distance equivalent.

Abstract

We show that given a lattice $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$, 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 $p_2(\Gamma)$. 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.