Buffon Discrepancy and the Steinhaus Longimeter

TL;DR

This paper investigates the distribution of one-dimensional sets within convex regions to minimize Buffon discrepancy, generalizing Steinhaus' construction and analyzing specific cases like the disk.

Contribution

It extends Steinhaus' method to construct sets with Buffon discrepancy bounds and demonstrates bounded discrepancy for the disk as length increases.

Findings

Existence of sets with Buffon discrepancy  L^{1/3}

Unit disk admits sets with uniformly bounded Buffon discrepancy

Generalization of Steinhaus construction for discrepancy bounds

Abstract

Let $\Omega \subset \mathbb{R}^2$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $\ell$ in $\mathbb{R}^2$ the number of intersections $\#(\ell \cap S)$ is proportional to the length $\mathcal{H}^1(\ell \cap \Omega)$ as much as possible; we use the term Buffon discrepancy for the largest error. A construction of Steinhaus can be generalized to prove the existence of sets with Buffon discrepancy $\lesssim L^{1/3}$. We also show that the unit disk $\mathbb{D}$ admits a set with uniformly bounded Buffon discrepancy as $L \rightarrow \infty$.