论文标题
$ U（1）$的最佳拓扑生成器
Optimal topological generators of $U(1)$
论文作者
论文摘要
萨尔纳克（Sarnak）的黄金平均猜想指出，所有整数$（m+1）\ le1+\ frac {2} {\ sqrt {5}} $对于所有整数$ m \ ge1 $，$φ$是$ 1 $ 1 $ 1 $ 1 $ 1 $+usip的$d_θ$是$d_θ$的$ 1 $+soip，共享此属性的值$θ$的$ \ MATHCAL {s} $，以及具有某些下限$ M \ ge m $的属性的人的集合$ \ Mathcal {t} $。值得注意的是，$ \ MATHCAL {S} \ text {mod} 1 $只有16个元素，而$ \ Mathcal {t} $是$ gl_2（\ Mathbb {z}）$ - $φ$的转换。
Sarnak's golden mean conjecture states that $(m+1)d_φ(m)\le1+\frac{2}{\sqrt{5}}$ for all integers $m\ge1$, where $φ$ is the golden mean and $d_θ$ is the discrepancy function for $m+1$ multiples of $θ$ modulo 1. In this paper, we characterize the set $\mathcal{S}$ of values $θ$ 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 $φ$.
