Polylogarithmic Full-Chord Buffon Discrepancy

Steinerberger introduced the Buffon discrepancy problem, asking how accurately a one-dimensional set of length $L$ in a convex body $\Omega$ can match the Crofton-predicted line-intersection counts, and proved an $O(L^{1/3})$ upper bound via a Steinhaus longimeter construction. His third open question asks whether restricting to sets built from full chords - intersections of lines with $\Omega$, the class containing every Steinhaus set - fundamentally changes the problem. It does. Using the Aistleitner-Bilyk-Nikolov star-discrepancy theorem for arbitrary measures, full-chord constructions with discrepancy $O\left((\log L)^{3/2}\right)$ are shown to exist for every compact convex body with finite piecewise $C^2$ boundary. In the disk, every full-chord construction is shown to have discrepancy at least $\Omega(\log L)$, via Schmidt's two-dimensional rectangle lower bound - where Steinerberger's concentric-circle construction, which is not full-chord, achieves discrepancy at most $100$.

This settles Steinerberger's third open question and separates the two models: in the disk, full chords cost you a factor growing like $\log L$ over what is achievable without the restriction. It also improves the Steinhaus-type $O(L^{1/3})$ to polylogarithmic within the full-chord class. It does not settle the Buffon discrepancy problem itself. Steinerberger's first question - whether every convex body admits a set of discrepancy $O(1)$, and if not what the truth is - is untouched, and the paper's closing line names it as the natural next question. Inside the full-chord model the order is pinned only between $\Omega(\log L)$ and $O\left((\log L)^{3/2}\right)$, and the lower bound is proved for the disk alone. The paper says the exponents are unlikely to be sharp. The upper bound is an existence statement: it inherits the Aistleitner-Bilyk-Nikolov theorem, which is proved by transference and supplies no explicit construction.