{
  "schema_version": 1,
  "problem_number": "AMR-129-0019",
  "title": "On Tabachnikov's Question on the Density of Reflected Wave Fronts: Balls and Ellipses",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Tabachnikov asked in 2026 whether, for a domain with smooth boundary in Euclidean space and a generic point P of it, the time-t wave front (the positions at time t of the unit-speed photons that leave P in all directions and are reflected optically at the boundary) becomes dense as t → ∞; he added that even the round disk is open. We use the definition of Kang and Knill: the front becomes dense if it meets every open ball for all sufficiently large t. We give a partial answer. In a ball of R^d, d ≥ 2, the front becomes dense for every source other than the centre, and for the centre it does not. For the disk this also follows from a theorem of Colin de Verdière and Vicente on the length of the front inside a subdomain (2020; Exp. Math. 2024), whose proof is a sketch and who do not state the density corollary; the ball reduces to the disk by rotational symmetry. We give a complete proof: on the invariant tori the motion is a linear flow whose frequencies satisfy a twist condition for every Fourier mode, and the Riemann–Lebesgue lemma gives weak convergence of the distribution of the position of a photon with a uniformly random direction to a limit of full support. In an ellipse with semi-major axis a the front becomes dense exactly when P is not a focus; from a focus all photons meet at a focus at the times 2a, 4a, …. Here the twist condition comes from elliptic coordinates and the limits of the frequencies at the ends of the two regimes of motion. We also recall, with an explicit C^∞ table of Penrose type, that for non-convex smooth domains the answer is negative in general; this is known in substance. General convex and generic smooth domains, the setting of the conjectures of Kang and Knill, remain open. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "wave fronts",
    "billiards",
    "dense wave fronts",
    "circular billiard",
    "elliptic billiard",
    "integrable billiards",
    "invariant tori",
    "twist condition",
    "oscillatory integrals",
    "illumination problem",
    "partial answer",
    "Open problems in billiards and quantitative symplectic geometry",
    "Tabachnikov",
    "Kang–Knill conjectures",
    "UnsolvedMath",
    "AMR-129-0019",
    "math.DS",
    "math.MG",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-129-0019/",
  "pdf_url": "https://eulersolve.org/papers/amr-129-0019/paper.pdf?v=b9bca1e9f6be",
  "doi": "10.5281/zenodo.23454031",
  "zenodo_record_url": "https://zenodo.org/records/23454031",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A partial answer. The disk case also follows from Theorem 4.1 of Colin de Verdière and Vicente (sketch proof; the density corollary is not stated there); general convex and generic smooth domains remain open. No priority is claimed.",
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    "source.zip": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}