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 $100$ .

Result: Proved(see note)
Status: Partial result
AI contribution: AI-assisted
Method: Argument
Field: Discrepancy theory / integral geometry
Posed by: Stefan Steinerberger
Year posed: 2026
Years open: 0y
Solved: 2026-05-18
Model: GPT-5.5
Vendor: OpenAI
Collaborators: Samuel Korsky
Verification: Unreviewed
Publication: Preprint
Significance: 5 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

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.

What the AI did

While the proof strategy (apply weighted versions of known discrepancy results in the necessary ways) was due to the author, GPT performed the bulk of the technical details and deserves a substantial amount of credit here.

Verification

A preprint by a single author, not refereed and not endorsed by anyone independent, so this stays Unreviewed.

This site checked the reduction the note rests on. Lemma 3.1 says the chords crossing a test line form a union of two rectangles in endpoint-pair space, of measure $2\mathcal{H}^1(\ell \cap \Omega)/\Lambda_\Omega$ - the identity that turns a Buffon problem into a two-dimensional rectangle discrepancy problem. It was confirmed exactly for the disk by quadrature at five arc widths (agreement to $10^{-9}$ ), and for an ellipse by sampling the kinematic measure in $(p,\theta)$ coordinates, which know nothing about endpoint pairs, giving agreement within 0.2% and an implied $\Lambda_\Omega$ of 4.6012 against a perimeter of 4.6026. The Aistleitner-Bilyk-Nikolov bound is quoted faithfully: their $(\log N)^{d-1/2}/N$ at $d=2$ is $(\log N)^{3/2}/N$ . An independent exact-supremum harness reproduces both known growth rates: $L^{0.289}$ for Steinhaus-type constructions against the proved $L^{1/3}$ , and $L^{0.511}$ for i.i.d. chords against the square root.

Neither theorem itself was checked. The upper bound rests on an existence result with no explicit construction, and the closest thing this site could build - a Halton set pushed through the Rosenblatt transform of $\mu_\Omega$ - fits $L^{0.346}$ , no better than Steinhaus. That is a limitation of the proxy, not evidence against the theorem. The $\Omega(\log L)$ lower bound is below the resolution of any feasible experiment.

Sources

Submitted by GoldenMongoose827 on 17 Aug 2026

Polylogarithmic Full-Chord Buffon Discrepancy

What was actually shown

What the AI did

Verification

Sources

Discussion