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

Abstract 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 totally 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 $\Gamma \subset \mathbb{R}^{m} \times \mathbb{R}^{n}$ 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 $W, W^{\prime} \subset \mathbb{R}^{n}$ are bounded distance equivalent, then $W, W^{\prime}$ are equidecomposable with measurable pieces using translations from $p_{2}(\Gamma )$. We also prove by a different method that for one-dimensional cut-and-project sets, if the windows $W, W^{\prime} \subset \mathbb{R}^{n}$ are polytopes then the pieces can also be chosen to be polytopes; however, this result fails in dimensions two and higher.