Bounded distance equivalence of cut-and-project sets and equidecomposability
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915776350388224 |
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| author | Grepstad, Sigrid |
| author_facet | Grepstad, Sigrid |
| contents | We show that given a lattice $Γ\subset \mathbb{R}^m \times \mathbb{R}^n$, and projections $p_1$ and $p_2$ onto $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively, cut-and-project sets obtained using Jordan measurable windows $W$ and $W'$ in $\mathbb{R}^n$ of equal measure are bounded distance equivalent only if $W$ and $W'$ are equidecomposable by translations in $p_2(Γ)$. As a consequence, we obtain an explicit description of the bounded distance equivalence classes in the hulls of simple quasicrystals. A corrigendum is appended at the end of the paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_05450 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounded distance equivalence of cut-and-project sets and equidecomposability Grepstad, Sigrid Dynamical Systems Number Theory 52C23, 52B45, 11K38 We show that given a lattice $Γ\subset \mathbb{R}^m \times \mathbb{R}^n$, and projections $p_1$ and $p_2$ onto $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively, cut-and-project sets obtained using Jordan measurable windows $W$ and $W'$ in $\mathbb{R}^n$ of equal measure are bounded distance equivalent only if $W$ and $W'$ are equidecomposable by translations in $p_2(Γ)$. As a consequence, we obtain an explicit description of the bounded distance equivalence classes in the hulls of simple quasicrystals. A corrigendum is appended at the end of the paper. |
| title | Bounded distance equivalence of cut-and-project sets and equidecomposability |
| topic | Dynamical Systems Number Theory 52C23, 52B45, 11K38 |
| url | https://arxiv.org/abs/2409.05450 |