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 $W, W' \subset \mathbb{R}^n$ 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; however this result fails in dimensions two and higher.
From: Nir Lev [view email]
[v1]
Wed, 26 Nov 2025 08:07:38 UTC (18 KB)
[v2]
Thu, 12 Feb 2026 16:13:42 UTC (22 KB)
[v3]
Tue, 23 Jun 2026 14:32:20 UTC (22 KB)
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