There have been many questions posted here about the sequence $s(x)_n = \sin(nx)$ and its corresponding series $S(x)$. In particular, it has been shown that $s(x)_n$ only converges if $x$ is a multiple of $\pi$ and moreover that its $\liminf$ and $\limsup$ for almost all $x$ are $1$ and $-1$, respectively. It follows immediately that the corresponding series does not converge if $x$ is not a multiple of $\pi$.
A related but separate questions is for which $x$ $\liminf S(x)$ and $\limsup S(x)$ are finite (treating the series $S(x)$ as a sequence of partial sums). Moreover, for which $x$ do these partial sums go between positive and negative infinitely often?
I am wondering to what degree one can draw a parallel between the terms $S(x)_n$ and a uniform one-dimensional random walk, which crosses 0 infinitely often but also goes arbitrarily far in the positive and negative directions.
We have: $$S_N(\alpha)\triangleq\sum_{n=0}^{N}\sin(n\alpha) = \frac{\cos\frac{\alpha}{2}-\cos\frac{(2N+1)\alpha}{2}}{2\sin\frac{\alpha}{2}}$$ (we can prove this identity by multiplying both sides by $2\sin\frac{\alpha}{2}$ and exploiting the fact that $2\sin\frac{\alpha}{2}\sin(n\alpha)=\cos((n-1/2)\alpha)-\cos((n+1/2)\alpha)$ leads to a telescopic sum) hence, if $\frac{\alpha}{\pi}$ is an irrational number, by the density of $e^{iN\alpha}$ in the unit circle we have: $$\limsup_{N}S_N(\alpha) = \frac{1}{2}\cot\frac{\alpha}{4},\qquad \liminf_{N}S_N(\alpha) = -\frac{1}{2}\tan\frac{\alpha}{4}$$ and since we are free to assume $\alpha\in(0,2\pi)$, then $S_N(\alpha)$ always oscillates between a rather large positive value and a rather small negative value.