Bounded common fundamental domains for two lattices
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908678109528064 |
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| author | Grepstad, Sigrid Kolountzakis, Mihail N. |
| author_facet | Grepstad, Sigrid Kolountzakis, Mihail N. |
| contents | We prove that for any two lattices $L, M \subseteq \mathbb{R}^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E \subseteq \mathbb{R}^d$ such that $E$ tiles $\mathbb{R}^d$ when translated by $L$ or by $M$. In fact, the set $E$ can be taken to be a finite union of polytopes. A consequence of this is that the indicator function of $E$ forms a Weyl--Heisenberg (Gabor) orthogonal basis of $L^2(\mathbb{R}^d)$ when translated by $L$ and modulated by $M^*$, the dual lattice of $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded common fundamental domains for two lattices Grepstad, Sigrid Kolountzakis, Mihail N. Classical Analysis and ODEs Metric Geometry 52B20, 52C22, 11H16 We prove that for any two lattices $L, M \subseteq \mathbb{R}^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E \subseteq \mathbb{R}^d$ such that $E$ tiles $\mathbb{R}^d$ when translated by $L$ or by $M$. In fact, the set $E$ can be taken to be a finite union of polytopes. A consequence of this is that the indicator function of $E$ forms a Weyl--Heisenberg (Gabor) orthogonal basis of $L^2(\mathbb{R}^d)$ when translated by $L$ and modulated by $M^*$, the dual lattice of $M$. |
| title | Bounded common fundamental domains for two lattices |
| topic | Classical Analysis and ODEs Metric Geometry 52B20, 52C22, 11H16 |
| url | https://arxiv.org/abs/2507.00604 |