OP-00515 Open

Asymptotic density of reflected wavefronts in a disk

Note: AIM — Open Applied Problems marks this solved (15 October 2026) by its own reduction to Theorem 3.9 of an article in Annales Henri Lebesgue (doi:10.5802/ahl.42), which AIM notes is not itself the result asked for. The reduction is in AIM's AI review and has not been checked here.

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Let $D = \{x \in \mathbb{R}^2 : |x| \le 1\}$ and $P \in \operatorname{int} D \setminus \{0\}$. Launch a unit-speed ray from $P$ in every direction, reflecting specularly at $\partial D$, and let $W_t(P)$ be the set of their positions at time $t$, the wave front at time $t$. From the centre the wave front returns periodically to the centre. Kang and Knill proved that on a flat $2$-torus every wave front becomes dense, in the sense of the limit below, and deduced the same for the billiard in a square.

Problem. Is $$\lim_{t \to \infty}\ \sup_{x \in D}\ \inf_{y \in W_t(P)} |x - y| = 0$$ for every such $P$? The limit concerns each sufficiently late wave front separately, not the union of the wave fronts over time.

Source: Kang and Knill (2026), §5.2, who expect density for every source other than the centre of a circular table; see also Tabachnikov, in Albach et al. (2026), for generic sources in smooth domains.

Imported from github.com/MColbrook/AIM.

Classification

Primary: DS — Dynamical Systems
Keywords: AIM Open Applied Problems, billiards, wave fronts

References

@article{kang2026,
  author        = {Kang, Emily and Knill, Oliver},
  title         = {Density of wave fronts},
  journal       = {Math. Intelligencer},
  year          = {2026},
  doi           = {10.1007/s00283-026-10528-z},
  eprint        = {2501.14611},
  archivePrefix = {arXiv},
}
@misc{albach2026,
  author        = {Albach, Bernhard and Barraud, Jean-Fran{\c{c}}ois and Bialy, Misha and Bimmermann, Johanna and Ch{\'a}vez C{\'a}liz, Ana and Damian, Mihai and Deschamps, Lina and Hryniewicz, Umberto and Humili{\`e}re, Vincent and Khesin, Boris and Maier, Levin and Moreno, Agustin and Oancea, Alexandru and Paris-Romaskevich, Olga and Sorrentino, Alfonso and Tabachnikov, Serge},
  title         = {Open problems in billiards and quantitative symplectic geometry},
  year          = {2026},
  eprint        = {2602.12896},
  archivePrefix = {arXiv},
}
@misc{aim-disk-wavefront-density,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Asymptotic density of reflected wavefronts in a disk}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/251-disk-wavefront-density.md},
  note         = {Entry 251 at commit a153f46, 2026-10-13},
}

Citing this problem

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OP-00515, "Asymptotic density of reflected wavefronts in a disk", clopen.solutions.