Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915794569396224 |
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| author | Etkind, Mark Mordechai Grepstad, Sigrid Kolountzakis, Mihail N. Lev, Nir |
| author_facet | Etkind, Mark Mordechai Grepstad, Sigrid Kolountzakis, Mihail N. Lev, Nir |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability Etkind, Mark Mordechai Grepstad, Sigrid Kolountzakis, Mihail N. Lev, Nir Metric Geometry Dynamical Systems 52C23, 52B45, 11K38 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. |
| title | Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability |
| topic | Metric Geometry Dynamical Systems 52C23, 52B45, 11K38 |
| url | https://arxiv.org/abs/2511.21148 |