Functional tilings and the Coven-Meyerowitz tiling conditions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908533874753536 |
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| author | Kiss, Gergely Londner, Itay Matolcsi, Máté Somlai, Gábor |
| author_facet | Kiss, Gergely Londner, Itay Matolcsi, Máté Somlai, Gábor |
| contents | Coven and Meyerowitz formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conjecture is reduced to sets which tile a finite cyclic group $\mathbb{Z}_M$. In this paper we consider a natural relaxation of this problem, where we replace sets with nonnegative functions $f,g$, such that $f(0)=g(0)=1$, $f\ast g=\mathbf{1}_{\mathbb{Z}_M}$ is a functional tiling, and $f, g$ satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Functional tilings and the Coven-Meyerowitz tiling conditions Kiss, Gergely Londner, Itay Matolcsi, Máté Somlai, Gábor Combinatorics Analysis of PDEs 05B45, 11B75, 20K01 (Primary) 11C08, 43A47, 51D20, 52C22 (Secondary) Coven and Meyerowitz formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conjecture is reduced to sets which tile a finite cyclic group $\mathbb{Z}_M$. In this paper we consider a natural relaxation of this problem, where we replace sets with nonnegative functions $f,g$, such that $f(0)=g(0)=1$, $f\ast g=\mathbf{1}_{\mathbb{Z}_M}$ is a functional tiling, and $f, g$ satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future. |
| title | Functional tilings and the Coven-Meyerowitz tiling conditions |
| topic | Combinatorics Analysis of PDEs 05B45, 11B75, 20K01 (Primary) 11C08, 43A47, 51D20, 52C22 (Secondary) |
| url | https://arxiv.org/abs/2411.03854 |