Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913110694035456 |
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| author | Korsky, Samuel |
| author_facet | Korsky, Samuel |
| contents | Let $Ω\subset \mathbb{R}^2$ be a bounded convex domain. Steinerberger (2026) introduced the Buffon discrepancy problem: given length $L$, construct a one-dimensional set $S\subsetΩ$ such that the number of intersections of $S$ with a line $\ell$ approximates the Crofton-normalized chord length $$ \frac{2L}{π|Ω|}\cdot\mathcal{H}^1(\ell\capΩ).$$ Steinerberger proved a universal upper bound of order $L^{1/3}$ using a Steinhaus longimeter construction, and showed that the disk admits bounded discrepancy. We prove that a randomly shifted Steinhaus construction improves the order of the universal upper bound to $L^{1/5}(\log L)^{2/5}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10096 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy Korsky, Samuel Combinatorics Metric Geometry Let $Ω\subset \mathbb{R}^2$ be a bounded convex domain. Steinerberger (2026) introduced the Buffon discrepancy problem: given length $L$, construct a one-dimensional set $S\subsetΩ$ such that the number of intersections of $S$ with a line $\ell$ approximates the Crofton-normalized chord length $$ \frac{2L}{π|Ω|}\cdot\mathcal{H}^1(\ell\capΩ).$$ Steinerberger proved a universal upper bound of order $L^{1/3}$ using a Steinhaus longimeter construction, and showed that the disk admits bounded discrepancy. We prove that a randomly shifted Steinhaus construction improves the order of the universal upper bound to $L^{1/5}(\log L)^{2/5}$. |
| title | Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2605.10096 |