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

TL;DR

This paper proves that certain bounded remainder sets and cut-and-project sets are equidecomposable with measurable pieces, using translations from specific groups, and explores conditions under which these pieces can be polytopes.

Contribution

It establishes measurable equidecomposability results for bounded remainder sets and cut-and-project sets, extending known geometric decompositions to higher dimensions and specific group actions.

Findings

Bounded remainder sets are equidecomposable via translations from a specific subgroup.

Bounded distance equivalent cut-and-project sets have equidecomposable windows.

Polytopal windows in 1D can be decomposed into polytopes, unlike in higher dimensions.

Abstract

We use the measurable Hall's theorem due to Cie\'sla 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 $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$ 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 $W, W' \subset \mathbb{R}^n$ are bounded distance equivalent, then $W, W'$ 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 $W, W' \subset \mathbb{R}^n$ are polytopes then the pieces can also be chosen to be polytopes; this fails in dimensions two and higher.