  Sarnak's golden mean conjecture states that
$(m+1)d_\varphi(m)\le1+\frac{2}{\sqrt{5}}$ for all integers $m\ge1$, where
$\varphi$ is the golden mean and $d_\theta$ is the discrepancy function for
$m+1$ multiples of $\theta$ modulo 1. In this paper, we characterize the set
$\mathcal{S}$ of values $\theta$ that share this property, as well as the set
$\mathcal{T}$ of those with the property for some lower bound $m\ge M$.
Remarkably, $\mathcal{S}\text{ mod }1$ has only 16 elements, whereas
$\mathcal{T}$ is the set of $GL_2(\mathbb{Z})$-transformations of $\varphi$.
