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

TL;DR

This work uses the measurable Hall's theorem to connect equidecomposability with discrepancy properties for two main settings: (i) bounded remainder sets in $\mathbb{R}^d$, where $A$ and $B$ of equal measure are shown to be equidecomposable with measurable pieces using translations from $\mathbb{Z}\alpha+\mathbb{Z}^d$; and (ii) higher-dimensional cut-and-project/model sets, where bounded distance equivalence of model sets implies equidecomposability of the defining windows via translations from $p_2(\Gamma)$. A key contribution is a gap-filled proof showing that for higher-rank lattices the windows must be equidecomposable up to measure zero if the corresponding model sets are BD-equivalent, with a separate one-dimensional argument giving stronger regularity on the pieces (Riemann measurable). The results illuminate deep links between discrepancy theory, equidecomposability, and the geometry of model sets, providing tools to construct and classify bounded remainder sets and to analyze BD-equivalence in the model-set framework. Overall, the paper extends known BRs and model-set correspondences to measurable equidecomposability, offering practical implications for tiling, discrepancy control, and a unified view of bounded distance phenomena.

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 given irrational $d$-dimensional vector $α$, then $A, B$ are equidecomposable with measurable pieces using translations from $\mathbb{Z} α+ \mathbb{Z}^d$; and (ii) given a lattice $Γ\subset \mathbb{R}^m \times \mathbb{R}^n$ 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 $p_2(Γ)$. We also prove by a different method that for one-dimensional cut-and-project sets the pieces can be chosen Riemann measurable.

Theorems & Definitions (22)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Definition 2.3: see CS22
• Theorem 2.4: CS22
• Theorem 3.1
• Theorem 4.1
• Lemma 4.2
• proof