Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy

TL;DR

This paper improves the upper bound on Buffon discrepancy for convex domains using a randomized Steinhaus construction, reducing the discrepancy order from L^{1/3} to L^{1/5} with a logarithmic factor.

Contribution

It introduces a randomized shift in the Steinhaus longimeter construction, achieving a better discrepancy bound for convex domains.

Findings

The discrepancy bound is improved to L^{1/5}( ext{log} L)^{2/5}.

Randomization enhances the universal upper bound for Buffon discrepancy.

The disk remains a special case with bounded 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 $L^{1/5}(\log L)^{2/5}$.