Optimal Point Sets Determining Few Distinct Angles

Abstract

We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2\le P(k)\le 6k$, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that $P(k)=\Theta (k)$ since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is well-understood.

arXiv comment10 pages, 6 figures. Revised version. Followup paper to arXiv:2206.04367 and arXiv:2108.12015