Buffon Discrepancy and the Steinhaus Longimeter
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912987622670336 |
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| author | Steinerberger, Stefan |
| author_facet | Steinerberger, Stefan |
| contents | Let $Ω\subset \mathbb{R}^2$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $\ell$ in $\mathbb{R}^2$ the number of intersections $\#(\ell \cap S)$ is proportional to the length $\mathcal{H}^1(\ell \cap Ω)$ as much as possible; we use the term Buffon discrepancy for the largest error. A construction of Steinhaus can be generalized to prove the existence of sets with Buffon discrepancy $\lesssim L^{1/3}$. We also show that the unit disk $\mathbb{D}$ admits a set with uniformly bounded Buffon discrepancy as $L \rightarrow \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27807 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Buffon Discrepancy and the Steinhaus Longimeter Steinerberger, Stefan Classical Analysis and ODEs Combinatorics Metric Geometry Let $Ω\subset \mathbb{R}^2$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $\ell$ in $\mathbb{R}^2$ the number of intersections $\#(\ell \cap S)$ is proportional to the length $\mathcal{H}^1(\ell \cap Ω)$ as much as possible; we use the term Buffon discrepancy for the largest error. A construction of Steinhaus can be generalized to prove the existence of sets with Buffon discrepancy $\lesssim L^{1/3}$. We also show that the unit disk $\mathbb{D}$ admits a set with uniformly bounded Buffon discrepancy as $L \rightarrow \infty$. |
| title | Buffon Discrepancy and the Steinhaus Longimeter |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2603.27807 |