Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability

We use the measurable Hall’s theorem due to Cieśla and Sabok to prove that (i) if two measurable sets $A,B\subset {\mathbb{R}}^d$ of the same measure are bounded remainder sets with respect to a given irrational $d$-dimensional vector $\alpha$, then $A,B$ are equidecomposable with measurable pieces using translations from $\mathbb{Z}\alpha +{\mathbb{Z}}^d$; and (ii) given a lattice $\mathrm{\Gamma }\subset {\mathbb{R}}^m\times {\mathbb{R}}^n$ with projections $p_1$ and $p_2$ onto ${\mathbb{R}}^m$ and ${\mathbb{R}}^n$ respectively, if two cut-and-project sets in ${\mathbb{R}}^m$ obtained from Riemann measurable windows $W,W^{\mathrm{\prime }}\subset {\mathbb{R}}^n$ are bounded distance equivalent, then $W,W^{\mathrm{\prime }}$ are equidecomposable with measurable pieces using translations from $p_2(\mathrm{\Gamma })$. We also prove by a different method that for one-dimensional cut-and-project sets, if the windows $W,W^{\mathrm{\prime }}\subset {\mathbb{R}}^n$ are polytopes then the pieces can also be chosen to be polytopes; this fails in dimensions two and higher.