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

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 $W^{\prime }$ in ${\mathbb{R}}^n$ of equal measure are bounded distance equivalent only if $W$ and $W^{\prime }$ 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.

arXiv commentThere is a gap in the proof of the main result (Theorem 1.1), and this carries over in the proofs of Corollaries 1.3, 1.4 and 1.7. A new and corrected proof of Theorem 1.1 is provided in arXiv:511.21148. Although the main result of the paper is correct as stated, Corollaries 1.3, 1.4, 1.7 and Conjecture 1.6 are provably wrong in dimensions greater than one