A proof of Sarnak's golden mean conjecture (Q2187117)

The golden ratio \(\theta^*=\frac{1+\sqrt{5}}{2}\) is common in mathematics, physics, art, architecture and nature. Let \(\theta \) be an irrational number and let \[ 0=a_0<a_1 <a_2 < \dots < a_m < a_{m+1}=1 \] be the sequence of points \(\{l\theta\}\), \(1\leq l \leq m \) (where \(\{x\}=x-[x] \) is the fractional part of \(x\)). Define \[ d_\theta(m)=\max\{(a_i-a_{i-1}),\ 1\leq i \leq m+1 \} \] and \[ \beta(\theta )=\sup_{m\geq 1} m d_\theta(m). \] Conjecture [Sarnak's conjecture]: For an arbitrary irrational number \(\theta\): \(\beta(\theta^*) \leq \beta(\theta) \). The following theorem is the main result of the article. Theorem. For the golden mean \(\theta^* \): \(\beta(\theta^* )=1+\frac{2\sqrt{5}}{2} \).