IVRII’S CONJECTURE FOR SOME CASES IN OUTER AND SYMPLECTIC BILLIARDS

Anastasiia Sharipova

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 empty interior for (2n+1,n) and (2n,n−1) outer billiard orbits by expressing dF as a shear (via dF = [-1 -2ρ/r; 0 -1]) and invoking a rotation–shear obstruction (Lemma 2.7), then applies a midpoint/parallelism correspondence for symplectic 3-cycles and a uniqueness-through-a-point argument for 4-cycles; these match the model’s strategy. The model reproduces the same product-of-rotations-and-positive-shears argument and the symplectic 3/4 steps. Two caveats: it misstated the shear coefficient (using κ instead of the radius ρ) and overclaimed "hence nowhere dense." The paper’s statements and proofs align with the correct dF formula and only claim empty interior, not nowhere density .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper delivers clear, self-contained proofs that the sets of (2n+1,n) and (2n,n−1) outer billiard orbits have empty interiors, and that 3- and 4-periodic symplectic orbits have empty interiors in all even dimensions. Its rotation–shear obstruction and geometric correspondences mirror established ideas but are presented succinctly and cleanly. Minor clarifications (explicit differential conventions, a brief justification of the higher-dimensional dimension obstruction) would strengthen the presentation without altering the results’ validity.