Optimal topological generators of $U(1)$

Abstract

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 $\varphi$.