Polylogarithmic Full-Chord Buffon Discrepancy

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.

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.