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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2504.00715 |
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| _version_ | 1866909775301705728 |
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| author | Trödler, Matěj Volec, Jan Vybíral, Jan |
| author_facet | Trödler, Matěj Volec, Jan Vybíral, Jan |
| contents | We elaborate on the intimate connection between the largest volume of an empty axis-parallel box in a set of $n$ points from $[0,1]^d$ and cover-free families from the extremal set theory. This connection was discovered in a recent paper of the authors. In this work, we apply a very recent result of Michel and Scott to obtain a whole range of new lower bounds on the number of points needed so that the largest volume of such a box is bounded by a given $\varepsilon$. Surprisingly, it turns out that for each of the new bounds, there is a choice of the parameters $d$ and $\varepsilon$ such that the bound outperforms the others. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower bounds on the minimal dispersion of point sets via cover-free families Trödler, Matěj Volec, Jan Vybíral, Jan Combinatorics We elaborate on the intimate connection between the largest volume of an empty axis-parallel box in a set of $n$ points from $[0,1]^d$ and cover-free families from the extremal set theory. This connection was discovered in a recent paper of the authors. In this work, we apply a very recent result of Michel and Scott to obtain a whole range of new lower bounds on the number of points needed so that the largest volume of such a box is bounded by a given $\varepsilon$. Surprisingly, it turns out that for each of the new bounds, there is a choice of the parameters $d$ and $\varepsilon$ such that the bound outperforms the others. |
| title | Lower bounds on the minimal dispersion of point sets via cover-free families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.00715 |