Bounded lattice tiles that pack with another lattice
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909803770544128 |
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| author | Grepstad, Sigrid Kolountzakis, Mihail N. Spyridakis, Emmanuil |
| author_facet | Grepstad, Sigrid Kolountzakis, Mihail N. Spyridakis, Emmanuil |
| contents | Suppose L and M are full-rank lattices in Euclidean space, such that vol(L) < vol(M). Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set F (we can even take F to be a finite union of polytopes) such that F+L is a tiling and F+M is a packing. If we do not require measurability of F it is often possible that a set F can be found tiling with both L and M even when L and M have different volumes, for instance if their intersection is trivial. We also show here that such a set can never be bounded if L and M have different volumes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07972 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded lattice tiles that pack with another lattice Grepstad, Sigrid Kolountzakis, Mihail N. Spyridakis, Emmanuil Classical Analysis and ODEs 52B20, 52C22, 11H16 Suppose L and M are full-rank lattices in Euclidean space, such that vol(L) < vol(M). Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set F (we can even take F to be a finite union of polytopes) such that F+L is a tiling and F+M is a packing. If we do not require measurability of F it is often possible that a set F can be found tiling with both L and M even when L and M have different volumes, for instance if their intersection is trivial. We also show here that such a set can never be bounded if L and M have different volumes. |
| title | Bounded lattice tiles that pack with another lattice |
| topic | Classical Analysis and ODEs 52B20, 52C22, 11H16 |
| url | https://arxiv.org/abs/2508.07972 |