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PreprintCC BY 4.0arXiv

Point sets determining few angles are almost contained in a line or circle

  • Krishnendu Bhowmick
  • Oliver Roche-Newton
  • Audie Warren
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Abstract

We prove a structural theorem for point sets in ℝ2 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 ∈ P which determines at least n1+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 ⊆ ℝ2 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.

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Cite

BibTeXRIS
BibTeX
@misc{bhowmick2026point,
  title = {{Point sets determining few angles are almost contained in a line or circle}},
  author = {Krishnendu Bhowmick and Oliver Roche-Newton and Audie Warren},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15270},
  eprint = {2609.15270},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15270},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Point sets determining few angles are almost contained in a line or circle
AU  - Krishnendu Bhowmick
AU  - Oliver Roche-Newton
AU  - Audie Warren
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15270
UR  - https://arxiv.org/abs/2609.15270
AB  - We prove a structural theorem for point sets in 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 ∈ 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 ⊆ 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.
ER  -