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Point sets determining few angles are almost contained in a line or circle — arXiv:2609.15270

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.15270
License (as archived): CC0-1.0
Archived: 2026-10-16T01:25:28.531Z
SHA-256 of the archived text: fe165b1de52371e2d2f92a4b8bff3bbefc10b79a45e306209671d1a0722635aa
Stable URL of this page: https://metraipolis.com/documento/dc_65bee97eba1e

How to cite

Point sets determining few angles are almost contained in a line or circle — arXiv:2609.15270 — arXiv (en), https://arxiv.org/abs/2609.15270 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_65bee97eba1e (Metraipolis archive dc_65bee97eba1e, sha256 fe165b1de52371e2d2f92a4b8bff3bbefc10b79a45e306209671d1a0722635aa)

BibTeX

@misc{dc65bee97eba1e,
  title = {Point sets determining few angles are almost contained in a line or circle — arXiv:2609.15270},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.15270},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_65bee97eba1e; sha256 fe165b1de52371e2d2f92a4b8bff3bbefc10b79a45e306209671d1a0722635aa},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.15270. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We prove a structural theorem for point sets in $\mathbb R^2$ which determine few pinned angles. More precisely, we prove the existence of an absolute constant $c>0$ such that if $n$ is sufficiently large and $P$ is a set of $n$ points then there exists a point $q \in P$ which determines at least $n^{1+c}$ distinct angles to other pairs of points of $P$, provided that $P$ is not of one of the following exceptional forms: all but at most one of the points of $P$ lie on a line; all but two points of $P$ lie on a line, and the two exceptional points are symmetric with respect to the line; all the points of $P$ lie on a circle; all but one of the points of $P$ lie on a circle, and the exceptional point is the centre of the circle. As a consequence, we answer a question of Corrádi, Erdős and Hajnal by showing that if $n$ is sufficiently large and $P \subseteq \mathbb R^2$ has cardinality $n$ and is not contained on a single line, then $P$ determines at least $n-2$ angles. Moreover, we prove that the unique point set attaining this minimum is the regular $n$-gon.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-15 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.15270 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T00:39:30.240Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-15?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).