Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy

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 $L^{1/5}(\log L{)}^{2/5}$.