Optimal topological generators of \(U(1)\) (Q2187119)

Let \(d_{\theta}(m)\) be a largest gap (modulo one) of \(m+1\) points \(0,\{\theta\},\{2\theta\},\ldots,\{m\theta\}\). Then it is proved that \((m+1)d_{\theta}(m)>1+\frac{2}{\sqrt{5}}\) for any sufficiently large \(m\) if and only if \(\theta\) is equivalent to the golden section. Further, assume \(D_M(\theta)=\max_{1\leq m \leq M} (m+1) d_{\theta}(m)\) and \(D(\theta)=\sup_M D_M(\theta)\). It was known previously that \(D(\theta)\geq 1+\frac{2}{\sqrt{5}}\). Now it is proved that there exist exactly 16 values \(\theta\) modulo one for which \(D(\theta)=1+\frac{2}{\sqrt{5}}\). Denote the set of such values by \(S\). Then for any \(\theta_0\in S\) there are infinitely many \(M\) such that \(\theta_0\in \arg\min_{\theta\in S} D_M(\theta)\).