Polylogarithmic Full-Chord Buffon Discrepancy

Abstract

Steinerberger introduced the Buffon discrepancy problem, asking how accurately a one-dimensional set of length $L$ in a convex body can match the Crofton-predicted line-intersection counts, and proved an $O\left(L^{1/3}\right)$ upper bound via a Steinhaus longimeter construction. Using the Aistleitner--Bilyk--Nikolov arbitrary-measure star-discrepancy theorem we demonstrate the existence of full-chord constructions with discrepancy $O\left((\log L{)}^{3/2}\right)$ for every fixed compact convex body with finite piecewise $C^2$ boundary. In the disk, we prove that every full-chord construction has discrepancy at least $\Omega \left(\log L\right)$, using Schmidt's two-dimensional rectangle discrepancy lower bound.