Repeated Angles in the Plane for Angles with Algebraic Tangents

We construct a set of points with $\mathrm{\Omega }(n^2\log n)$ triples determining an angle $\theta$ whenever $\tan (\theta )$ is algebraic over $\mathbb{Q}$, matching the upper bound of Pach and Sharir. This improves upon the original construction, which was optimal only for $\tan (\theta )=a\sqrt{m}\mathrm{/}b$ with $a,b,m$ positive integers.