On the minimal period of integer tilings
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910512363601920 |
|---|---|
| author | Łaba, Izabella Zakharov, Dmitrii |
| author_facet | Łaba, Izabella Zakharov, Dmitrii |
| contents | If a finite set $A$ tiles the integers by translations, it also admits a tiling whose period $M$ has the same prime factors as $|A|$. We prove that the minimal period of such a tiling is bounded by $\exp(c(\log D)^2/\log\log D)$, where $D$ is the diameter of $A$. In the converse direction, given $ε>0$, we construct tilings whose minimal period has the same prime factors as $|A|$ and is bounded from below by $D^{3/2-ε}$. We also discuss the relationship between minimal tiling period estimates and the Coven-Meyerowitz conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14824 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the minimal period of integer tilings Łaba, Izabella Zakharov, Dmitrii Number Theory Combinatorics 05B45, 52C22 If a finite set $A$ tiles the integers by translations, it also admits a tiling whose period $M$ has the same prime factors as $|A|$. We prove that the minimal period of such a tiling is bounded by $\exp(c(\log D)^2/\log\log D)$, where $D$ is the diameter of $A$. In the converse direction, given $ε>0$, we construct tilings whose minimal period has the same prime factors as $|A|$ and is bounded from below by $D^{3/2-ε}$. We also discuss the relationship between minimal tiling period estimates and the Coven-Meyerowitz conjecture. |
| title | On the minimal period of integer tilings |
| topic | Number Theory Combinatorics 05B45, 52C22 |
| url | https://arxiv.org/abs/2406.14824 |