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arXiv · 2605.10096

Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy

Samuel Korsky
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Abstract

Let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain. Steinerberger (2026) introduced the Buffon discrepancy problem: given length $L$, construct a one-dimensional set $S\subset\Omega$ such that the number of intersections of $S$ with a line $\ell$ approximates the Crofton-normalized chord length $$ \frac{2L}{\pi|\Omega|}\cdot\mathcal{H}^1(\ell\cap\Omega).$$ 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 $L^{1/5}(\log L)^{2/5}$.

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Samuel Korsky. 2026-05-11. Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy. https://arxiv.org/abs/2605.10096

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