Geometric implications of weak tiling
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911157019738112 |
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| author | Kolountzakis, Mihail N. Lev, Nir Matolcsi, Máté |
| author_facet | Kolountzakis, Mihail N. Lev, Nir Matolcsi, Máté |
| contents | The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling and establish some geometric properties of sets that weakly tile their complement. If $A \subset \mathbb{R}^d$ is a convex polytope, we give a direct and self-contained proof that $A$ must be symmetric and have symmetric facets. If $A \subset \mathbb{R}$ is a finite union of intervals, we give a necessary condition on the lengths of the gaps between the intervals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23631 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric implications of weak tiling Kolountzakis, Mihail N. Lev, Nir Matolcsi, Máté Classical Analysis and ODEs 42B10, 42C05, 52C22 The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling and establish some geometric properties of sets that weakly tile their complement. If $A \subset \mathbb{R}^d$ is a convex polytope, we give a direct and self-contained proof that $A$ must be symmetric and have symmetric facets. If $A \subset \mathbb{R}$ is a finite union of intervals, we give a necessary condition on the lengths of the gaps between the intervals. |
| title | Geometric implications of weak tiling |
| topic | Classical Analysis and ODEs 42B10, 42C05, 52C22 |
| url | https://arxiv.org/abs/2506.23631 |