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 $α$, 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, 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.

💡 Research Summary

The paper investigates two classical objects—bounded remainder sets (BRS) and cut‑and‑project (C&P) sets—from the perspective of measurable equidecomposability. The authors’ main technical tool is the “measurable Hall’s theorem” proved by Cieśla and Sabok (2022), which asserts that for a free probability‑measure‑preserving (pmp) action of a finitely generated abelian group G on a standard Borel probability space, any two measurable G‑uniform sets that satisfy Hall’s condition almost everywhere admit a G‑equidecomposition with measurable pieces.

Bounded remainder sets.
Let α∈ℝᵈ be such that 1,α₁,…,α_d are rationally independent. A measurable set A⊂ℝᵈ is a BRS if the discrepancy
 supₙ |∑_{k=0}^{n‑1}χ_A(x+kα)−n·mes A|
is bounded uniformly in n for almost every x. The long‑standing question (raised in Grepstad–Larcher 2015) asked whether two BRS of equal measure must be equidecomposable using only translations from the dense subgroup ℤα+ℤᵈ, and whether the pieces can be taken measurable. The authors answer affirmatively for all BRS, not only Riemann‑measurable or polyhedral ones.

The proof proceeds by embedding the problem into the product space X = 𝕋ᵈ×ℤ_q, where q is a large integer covering both A and B by q translates of the unit cube. Define the group G = ℤ×ℤ_q acting by
 (n,σ)·(x,τ) = (x+nα, σ+τ).
Lift A and B to measurable subsets A′, B′⊂X. The BRS discrepancy bound yields, for almost every x∈𝕋ᵈ, the Hall inequality on each G‑orbit: the number of points of A′ in a finite segment of the orbit never exceeds the number of points of B′ in the same segment by more than a fixed constant C, and vice‑versa. Hence A′ and B′ satisfy Hall’s condition a.e. Since A′, B′ are G‑uniform, the measurable Hall theorem provides a G‑equidecomposition with measurable pieces. Projecting back to ℝᵈ gives the desired equidecomposition of A and B using translations from ℤα+ℤᵈ. Consequently, any two BRS of the same volume are measurably equidecomposable, and the construction works for arbitrary measurable BRS.

Cut‑and‑project sets.
Consider a lattice Γ⊂ℝᵐ×ℝⁿ and its coordinate projections p₁:ℝᵐ×ℝⁿ→ℝᵐ, p₂:ℝᵐ×ℝⁿ→ℝⁿ. For a Riemann‑measurable window W⊂ℝⁿ, the associated C&P set is
 Λ(W)=p₁(Γ∩(ℝᵐ×W)).
Two such sets Λ(W) and Λ(W′) are bounded‑distance equivalent if there exists a bijection φ:Λ(W)→Λ(W′) with sup_{x∈Λ(W)}‖x−φ(x)‖<∞. The authors show that bounded‑distance equivalence forces the windows W and W′ to satisfy Hall’s condition almost everywhere with respect to the translation group G = p₂(Γ). The argument mirrors the BRS case: for each G‑orbit (i.e., each fiber of p₁), the discrepancy between the numbers of points contributed by W and by W′ is uniformly bounded because of the bounded‑distance bijection. Hence Hall’s condition holds a.e., and the measurable Hall theorem yields a p₂(Γ)‑equidecomposition of W and W′ with measurable pieces (up to a null set). This fills a gap left in Grepstad’s earlier work and provides a clean measurable version of the well‑known fact that bounded‑distance equivalence of model sets implies “shape” equivalence of their windows.

One‑dimensional model sets and polyhedral pieces.
In the special case m=1 (so the physical space is a line), the authors give an alternative, more constructive proof that does not rely on the measurable Hall theorem. By ordering points along the line and using the bounded‑distance bijection, they produce an explicit matching that respects the order. When the windows W and W′ are finite unions of intervals (i.e., polytopes in ℝ¹), the matching can be arranged so that each piece of the equidecomposition is itself a finite union of intervals. Thus, in one dimension, polyhedral windows lead to polyhedral equidecomposition pieces.

However, the authors also demonstrate that this polyhedral refinement fails in higher dimensions. They construct explicit examples of polyhedral windows in ℝ² (and higher) whose associated model sets are bounded‑distance equivalent, yet any measurable equidecomposition of the windows necessarily involves non‑polyhedral pieces. This highlights a genuine geometric obstruction that appears only when the internal space has dimension ≥2.

Structure of the paper.
Section 2 reviews equidecomposability, Hall’s condition, and states the measurable Hall theorem. Section 3 develops the BRS result, including the embedding into 𝕋ᵈ×ℤ_q and the verification of Hall’s condition using the classical discrepancy bound. Section 4 treats cut‑and‑project sets, establishing the Hall condition from bounded‑distance equivalence and applying the measurable Hall theorem. Section 5 presents the alternative one‑dimensional argument, proves the polyhedral refinement in dimension 1, and supplies counterexamples for dimensions ≥2. An appendix clarifies the status of results announced in the retracted Grepstad