Absolute Summation

$\begin{aligned} S & = \lim_{n \to \infty} \sum_{k=0}^n \left| \frac {2 \pi \cos (k \pi (3 - \sqrt 5))}n \right| \\ & = \lim_{n \to \infty} 2 \pi \sum_{k=0}^n \frac {\left|\cos (3 k \pi - \sqrt 5 k \pi ) \right|}n \\ & = \lim_{n \to \infty} 2 \pi \sum_{k=0}^n \frac {\left|{\color{#3D99F6}\cos (3 k \pi)} \cos(\sqrt 5 k \pi) + {\color{#D61F06}\sin (3 k \pi)} \sin (\sqrt 5 k \pi)\right|}n \\ & = \lim_{n \to \infty} 2 \pi \sum_{k=0}^n \frac {\left|{\color{#3D99F6} (\pm 1)} \cos(\sqrt 5 k \pi) + {\color{#D61F06}(0)} \sin (\sqrt 5 k \pi)\right|}n \\ & = \lim_{n \to \infty} 2 \pi \color{#3D99F6} \sum_{k=0}^n \frac {\left| \cos(\sqrt 5 k \pi) \right|}n \quad \quad \small \text{By equidistribution theorem} \\ & = 2 \pi \color{#3D99F6} \int_{-\frac \pi 2}^\frac \pi 2 \frac {\cos x}\pi dx \\ & = 2 \sin x \big|_{-\frac \pi 2}^\frac \pi 2 \\ & = \boxed{4} \end{aligned}$

Reference: Equidistribution theorem