Polylogarithmic Full-Chord Buffon Discrepancy

Samuel Korsky

arxiv: 2605.23020 · v1 · pith:DDFRISKCnew · submitted 2026-05-21 · 🧮 math.CO · math.MG

Polylogarithmic Full-Chord Buffon Discrepancy

Samuel Korsky This is my paper

Pith reviewed 2026-05-25 05:24 UTC · model grok-4.3

classification 🧮 math.CO math.MG

keywords Buffon discrepancyfull-chord constructionsstar discrepancyconvex bodiespolylogarithmic boundsCrofton formula

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The pith

Full-chord constructions achieve O((log L)^{3/2}) Buffon discrepancy for any fixed compact convex body with piecewise C^2 boundary.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that full-chord constructions of total length L exist whose Buffon discrepancy is bounded by O((log L)^{3/2}). This improves the earlier O(L^{1/3}) upper bound obtained from Steinhaus longimeter constructions. The argument applies the Aistleitner--Bilyk--Nikolov star-discrepancy result to the measures generated by the chords. For the disk the same constructions cannot do better than Omega(log L) by a reduction to Schmidt's rectangle discrepancy lower bound. A reader would care because the result replaces a polynomial dependence on L with a much slower polylogarithmic one while keeping the construction geometrically simple.

Core claim

Using the Aistleitner--Bilyk--Nikolov arbitrary-measure star-discrepancy theorem we demonstrate the existence of full-chord constructions with discrepancy O((log L)^{3/2}) for every fixed compact convex body with finite piecewise C^2 boundary. In the disk, every full-chord construction has discrepancy at least Omega(log L) by Schmidt's two-dimensional rectangle discrepancy lower bound.

What carries the argument

Application of the Aistleitner--Bilyk--Nikolov star-discrepancy theorem to the line-intersection measures induced by full-chord sets.

If this is right

Buffon discrepancy improves from polynomial in L to polylogarithmic for the stated class of bodies.
The same polylogarithmic upper bound holds uniformly across all fixed compact convex bodies with the given boundary regularity.
In the disk the gap between the Omega(log L) lower bound and the O((log L)^{3/2}) upper bound is only a square-root log factor.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may transfer to other integral-geometric discrepancy settings that reduce to star discrepancy of induced measures.
Full-chord sets appear nearly optimal for the Buffon matching task once the body is fixed.
Similar polylogarithmic constructions could be sought in higher-dimensional analogues of the Crofton formula.

Load-bearing premise

The Aistleitner--Bilyk--Nikolov star-discrepancy theorem applies directly to the measures induced by full-chord constructions without additional restrictions or modifications.

What would settle it

An explicit full-chord construction in a convex body whose measured discrepancy grows faster than any polylogarithmic function of L, or a demonstration that the star-discrepancy theorem cannot be invoked on these particular measures.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies an external star-discrepancy theorem to lift the Buffon upper bound from polynomial to polylogarithmic and adds a log lower bound in the disk.

read the letter

This paper improves the upper bound on Buffon discrepancy for full-chord constructions from O(L^{1/3}) to O((log L)^{3/2}) across compact convex bodies with piecewise C^2 boundary, and shows an Omega(log L) lower bound specifically in the disk. The advance comes from invoking the Aistleitner-Bilyk-Nikolov result on arbitrary measures rather than building a new construction from scratch like the earlier Steinhaus longimeter approach. That is the concrete gain: a cleaner exponent that holds uniformly for the stated class of bodies. The lower bound via Schmidt's rectangle discrepancy is a useful complement that pins down the order in the disk case. The citation pattern is straightforward and appropriate, resting on established discrepancy tools without circularity. The main soft spot is the realization step: the external theorem supplies a low-discrepancy measure on the space of lines, but full-chord sets only generate measures whose support and density come from complete chords inside the body. If the paper supplies a discretization or approximation argument that keeps the error inside the polylog term for every such boundary, the claim holds; otherwise the bound could pick up extra factors. The abstract presents the result as direct, so the manuscript presumably controls this, but that is the part worth checking line by line. Readers working in geometric discrepancy or integral geometry will find the application useful. The work is coherent on its own terms and shows clear engagement with the prior literature, so it deserves a serious referee even if the realization details need tightening.

Referee Report

1 major / 0 minor

Summary. The paper claims an improved upper bound of O((log L)^{3/2}) on the Buffon discrepancy for full-chord constructions of total length L inside any fixed compact convex body with finite piecewise C² boundary, obtained by applying the Aistleitner--Bilyk--Nikolov arbitrary-measure star-discrepancy theorem; it also establishes a lower bound of Ω(log L) in the disk by reduction to Schmidt's two-dimensional rectangle discrepancy result. This improves Steinerberger's earlier O(L^{1/3}) construction.

Significance. If the application of the external theorem is valid without uncontrolled remainder terms arising from the full-chord restriction, the result would establish that polylogarithmic discrepancy is achievable within the restricted class of full-chord measures, representing a substantial quantitative advance over the previous power-law bound. The matching lower bound (up to the precise exponent) in the disk supplies a useful complement and indicates near-optimality in that case.

major comments (1)

[proof of the main upper-bound result (invocation of Aistleitner--Bilyk--Nikolov theorem)] The central existence claim rests on the Aistleitner--Bilyk--Nikolov theorem furnishing a measure that can be realized (or approximated with error o((log L)^{3/2})) by the intersection measure of a union of full chords; the manuscript provides no explicit construction, discretization, or error-control argument showing that the theorem's output lies in (or is close to) the class of measures inducible by full chords for arbitrary piecewise-C² boundaries. This step is load-bearing for the stated exponent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance, and for identifying this key point in the proof. We address the major comment below.

read point-by-point responses

Referee: [proof of the main upper-bound result (invocation of Aistleitner--Bilyk--Nikolov theorem)] The central existence claim rests on the Aistleitner--Bilyk--Nikolov theorem furnishing a measure that can be realized (or approximated with error o((log L)^{3/2})) by the intersection measure of a union of full chords; the manuscript provides no explicit construction, discretization, or error-control argument showing that the theorem's output lies in (or is close to) the class of measures inducible by full chords for arbitrary piecewise-C² boundaries. This step is load-bearing for the stated exponent.

Authors: We agree that the current manuscript lacks an explicit discretization or error-control argument connecting the Aistleitner--Bilyk--Nikolov output measure to the class of full-chord inducible measures, and that this is a load-bearing step. In the revised version we will add a dedicated subsection (or short appendix) that (i) recalls the relevant density result for chord measures in the space of measures on a piecewise-C² convex body, (ii) constructs an explicit finite union of full chords whose induced measure approximates the ABN measure in total variation (or in the dual norm relevant to Buffon discrepancy), and (iii) verifies that the resulting approximation error is o((log L)^{3/2}) uniformly for any fixed body with the stated boundary regularity. This will make the invocation of the external theorem fully rigorous while preserving the claimed exponent. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external theorems

full rationale

The paper derives its O((log L)^{3/2}) upper bound by invoking the Aistleitner--Bilyk--Nikolov arbitrary-measure star-discrepancy theorem (distinct authors) and its Omega(log L) lower bound via Schmidt's rectangle discrepancy result. No derivation step reduces by construction to a self-definition, a fitted input relabeled as a prediction, or a load-bearing self-citation chain; the central existence claim for full-chord constructions is presented as following from the external theorem's application rather than from any internal loop or renaming of known patterns. The derivation is therefore independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the applicability of two external discrepancy theorems; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption The Aistleitner--Bilyk--Nikolov arbitrary-measure star-discrepancy theorem holds and applies to Buffon measures from full-chord sets.
Directly invoked to obtain the O((log L)^{3/2}) upper bound.
domain assumption Schmidt's two-dimensional rectangle discrepancy lower bound applies to the disk Buffon setting.
Invoked to obtain the Ω(log L) lower bound.

pith-pipeline@v0.9.0 · 5653 in / 1384 out tokens · 31143 ms · 2026-05-25T05:24:22.758558+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 2 internal anchors

[1]

Steinerberger, Buffon discrepancy and the Steinhaus longimeter, arXiv:2603.27807, 2026

S. Steinerberger, Buffon discrepancy and the Steinhaus longimeter, arXiv:2603.27807, 2026

work page arXiv 2026
[2]

Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy

S. Korsky, Randomly shifted Steinhaus longimeters and Buffon discrepancy, arXiv:2605.10096, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[3]

Tusn\'ady's problem, the transference principle, and non-uniform QMC sampling

C. Aistleitner, D. Bilyk, and A. Nikolov, Tusn´ ady’s problem, the transference principle, and non-uniform QMC sampling, arXiv:1703.06127

work page internal anchor Pith review Pith/arXiv arXiv
[4]

W. M. Schmidt, Irregularities of distribution. VII,Acta Arith.21 (1972), 45–50. 11

work page 1972

[1] [1]

Steinerberger, Buffon discrepancy and the Steinhaus longimeter, arXiv:2603.27807, 2026

S. Steinerberger, Buffon discrepancy and the Steinhaus longimeter, arXiv:2603.27807, 2026

work page arXiv 2026

[2] [2]

Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy

S. Korsky, Randomly shifted Steinhaus longimeters and Buffon discrepancy, arXiv:2605.10096, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[3] [3]

Tusn\'ady's problem, the transference principle, and non-uniform QMC sampling

C. Aistleitner, D. Bilyk, and A. Nikolov, Tusn´ ady’s problem, the transference principle, and non-uniform QMC sampling, arXiv:1703.06127

work page internal anchor Pith review Pith/arXiv arXiv

[4] [4]

W. M. Schmidt, Irregularities of distribution. VII,Acta Arith.21 (1972), 45–50. 11

work page 1972