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

We show that given a lattice $\mathrm{\Gamma }\subset {\mathbb{R}}^m\times {\mathbb{R}}^n$, 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 $W^{\mathrm{\prime }}$ in ${\mathbb{R}}^n$ of equal measure are bounded distance equivalent only if $W$ and $W^{\mathrm{\prime }}$ are equidecomposable by translations in $p_2(\mathrm{\Gamma })$. 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.