Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded convex domain. Steinerberger (2026) introduced the Buffon discrepancy problem: given length $L$, construct a one-dimensional set $S\subset \mathrm{\Omega }$ such that the number of intersections of $S$ with a line $\mathrm{\ell }$ approximates the Crofton-normalized chord length $$\frac{2L}{\pi \mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}\cdot {\mathcal{H}}^1(\mathrm{\ell }\cap \mathrm{\Omega })\mathrm{.}$$ Steinerberger proved a universal upper bound of order $L^{1\mathrm{/}3}$ 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 $L^{1\mathrm{/}5}(\log L{)}^{2\mathrm{/}5}$.