New upper bound for lattice covering by spheres
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908482532278272 |
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| author | Gao, Jun Liu, Xizhi Pikhurko, Oleg Sun, Shumin |
| author_facet | Gao, Jun Liu, Xizhi Pikhurko, Oleg Sun, Shumin |
| contents | We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^β n \big)$ as $n\to\infty$, where \begin{align*}
β:= \frac{1}{2} \log_2 \left(\frac{8 π\mathrm{e}}{3\sqrt 3}\right)=1.85837...\,. \end{align*} This improves upon the previously best known upper bound by Rogers from 1959 of $O\big(n \ln^α n \big)$, where $α:= \frac{1}{2} \log_{2}(2π\mathrm{e})=2.0471...\,.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_06446 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New upper bound for lattice covering by spheres Gao, Jun Liu, Xizhi Pikhurko, Oleg Sun, Shumin Metric Geometry Combinatorics We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^β n \big)$ as $n\to\infty$, where \begin{align*} β:= \frac{1}{2} \log_2 \left(\frac{8 π\mathrm{e}}{3\sqrt 3}\right)=1.85837...\,. \end{align*} This improves upon the previously best known upper bound by Rogers from 1959 of $O\big(n \ln^α n \big)$, where $α:= \frac{1}{2} \log_{2}(2π\mathrm{e})=2.0471...\,.$ |
| title | New upper bound for lattice covering by spheres |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2508.06446 |