AN IMPROVED UPPER BOUND ON THE NUMBER OF BILLIARD BALL COLLISIONS

Krzysztof Burdzy

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s main theorem and proof are internally consistent and correctly yield the bound 800 (1300·32^{5d})^n n^{((3/2)·5d+3)n+2}, with clear dependence on the short-time BFK estimate and the BD20 quantitative splitting plus a carefully counted branching scheme. By contrast, the candidate solution conflates constants and steps: (i) it misstates the unit-interval wall-count bound (using 6^d and then replacing it by 32^{5d}, while the paper derives 5^d), (ii) it attributes a bound of the form 800(1300·W)^n n^{3n+2} to BFK — that bound does not appear in the paper and mixes constants from later leaf-counting, and (iii) it injects a dimension-dependent factor 5d into the number of leaves, whereas the paper’s leaf-count is independent of d and equals 1300^n n^{3n}. Although the candidate ends with the correct final expression, the intermediate steps are incorrect or unjustified.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper gives a sharp and explicit universal upper bound of order exp{c(d) n log n} for the number of hard-ball collisions, significantly improving prior universal bounds. The proof is conceptually clean, combining a modern quantitative splitting theorem with classic BFK estimates, and carefully tracks constants through a branching argument. Minor clarifications to constant origins and a brief geometric aside for the 5\^d neighbor bound would make it even more accessible.