2403.19954
BILLIARDS IN POLYHEDRA: A METHOD TO CONVERT 2-DIMENSIONAL UNIFORMITY TO 3-DIMENSIONAL UNIFORMITY
J. Beck, W.W.L. Chen, Y. Yang
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves uniform distribution in rational polygonal right prisms by projecting to the planar translation surface and applying a quantitative transference theorem for arithmetic progressions, together with the KMS almost-every-direction unique ergodicity on translation surfaces. The candidate solution proves the same statement via unfolding, product unique ergodicity (horizontal KMS + vertical irrational rotation), and a covering/pushforward argument. Both yield the same conclusion for almost every pair (starting point, direction); the proofs are substantively different but compatible with each other and with the paper’s statements .
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The manuscript provides a clear conversion principle from 2D uniform distribution on rational polygons to 3D uniform distribution in rational polygonal right prisms, using an arithmetic-progression sampling method. The result is correct and of interest to the polygonal/translation-surface community. Some expository refinements (clarifying the last implication from sampled uniformity on P to uniformity in M, consolidating notation, and explicitly remarking about the measure-zero set of vertical rational directions) would further improve readability.