The exact linear bound conjecture for few-angle point sets

Let $P(k)$ denote the minimum number of distinct angles determined by a finite set of points in the plane whose cardinality is at least $k$.

Exact linear bound conjecture. The lower bound on $P(k)$ in Theorem is tight. Namely,

$$P(2k)=2k+3\quad\text{and}\quad P(2k+1)=2k+3$$

for all $k\geq 1$.

The conjecture proposes exact values for $P(k)$ in every parity class, strengthening the currently established lower bound. It concerns planar point sets; investigating $P(k)$ for point sets in more than two dimensions remains an open problem.