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 $\hat{I}\pm$, then $A,B$ are equidecomposable with measurable pieces using translations from $\mathbb{Z}\hat{I}\pm +{\mathbb{Z}}^d$; and (ii) given a lattice $\hat{I}“\subset {\mathbb{R}}^m\times {\mathbb{R}}^n$ 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 $p_2(\hat{I}“)$. 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.

To appear in International Mathematics Research Notices IMRN