Bounded and measurable common fundamental domains for two lattices
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915821242023936 |
|---|---|
| author | Spyridakis, Emmanouil |
| author_facet | Spyridakis, Emmanouil |
| contents | Suppose that $L, M$ are two full-rank lattices in Euclidean space with $\text{vol}(L)=\text{vol}(M)$. We give a new proof on the existence of a bounded and Lebesgue measurable set that tiles $\mathbb{R}^d$ with both $L,M$ using the measurable Hall's Theorem which was proved by T.Ciésla and M. Sabok. This proof is direct and does not go through the intermediate results on cut-and-project sets involved in the proof given by S.Grepstad and M.Kolountzakis. We also show the existence of a bounded, set-theoretic (i.e., not necessarily measurable) common fundamental domain of $L,M$ assuming only that $\text{vol}(L)=\text{vol}(M)$. Combining these results we show the existence of a bounded and Lebesgue measurable common fundamental domain for any two full-rank lattices of equal volumes. Finally we show that a set-theoretic bounded, common fundamental domain cannot exist when $\text{vol}(L)\neq \text{vol}(M)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23480 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bounded and measurable common fundamental domains for two lattices Spyridakis, Emmanouil Classical Analysis and ODEs Metric Geometry 52B20, 52C22, 11H16 Suppose that $L, M$ are two full-rank lattices in Euclidean space with $\text{vol}(L)=\text{vol}(M)$. We give a new proof on the existence of a bounded and Lebesgue measurable set that tiles $\mathbb{R}^d$ with both $L,M$ using the measurable Hall's Theorem which was proved by T.Ciésla and M. Sabok. This proof is direct and does not go through the intermediate results on cut-and-project sets involved in the proof given by S.Grepstad and M.Kolountzakis. We also show the existence of a bounded, set-theoretic (i.e., not necessarily measurable) common fundamental domain of $L,M$ assuming only that $\text{vol}(L)=\text{vol}(M)$. Combining these results we show the existence of a bounded and Lebesgue measurable common fundamental domain for any two full-rank lattices of equal volumes. Finally we show that a set-theoretic bounded, common fundamental domain cannot exist when $\text{vol}(L)\neq \text{vol}(M)$. |
| title | Bounded and measurable common fundamental domains for two lattices |
| topic | Classical Analysis and ODEs Metric Geometry 52B20, 52C22, 11H16 |
| url | https://arxiv.org/abs/2602.23480 |