Polylogarithmic Full-Chord Buffon Discrepancy
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917522705481728 |
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| author | Korsky, Samuel |
| author_facet | Korsky, Samuel |
| contents | Steinerberger introduced the Buffon discrepancy problem, asking how accurately a one-dimensional set of length $L$ in a convex body can match the Crofton-predicted line-intersection counts, and proved an $O\left(L^{1/3}\right)$ upper bound via a Steinhaus longimeter construction. Using the Aistleitner--Bilyk--Nikolov arbitrary-measure star-discrepancy theorem we demonstrate the existence of full-chord constructions with discrepancy $O\left((\log L)^{3/2}\right)$ for every fixed compact convex body with finite piecewise $C^2$ boundary. In the disk, we prove that every full-chord construction has discrepancy at least $Ω\left(\log L\right)$, using Schmidt's two-dimensional rectangle discrepancy lower bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23020 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polylogarithmic Full-Chord Buffon Discrepancy Korsky, Samuel Combinatorics Metric Geometry Steinerberger introduced the Buffon discrepancy problem, asking how accurately a one-dimensional set of length $L$ in a convex body can match the Crofton-predicted line-intersection counts, and proved an $O\left(L^{1/3}\right)$ upper bound via a Steinhaus longimeter construction. Using the Aistleitner--Bilyk--Nikolov arbitrary-measure star-discrepancy theorem we demonstrate the existence of full-chord constructions with discrepancy $O\left((\log L)^{3/2}\right)$ for every fixed compact convex body with finite piecewise $C^2$ boundary. In the disk, we prove that every full-chord construction has discrepancy at least $Ω\left(\log L\right)$, using Schmidt's two-dimensional rectangle discrepancy lower bound. |
| title | Polylogarithmic Full-Chord Buffon Discrepancy |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2605.23020 |