See also Theorem 8.3.1 in Dajani and Kalle's book A first course in ergodic theory routledge.com/A-First-Course-in-Ergodic-Theory/Dajani-Kalle/‌​p/…

The answer is positive if $\mathbb{C}^n$ is endowed with the canonical hermitian norm

I do not know whether the answer is known. For example, a similar question, the equirepartition modulo 1 of $((3/2)^n)_{n \ge 0}$ is an open question (to my knowledge). Indicating where the questions come from would be useful: is it related to a question in research or does it come from an exercise?

First, you need to take a continuous version of $X$ - which is known to exist here - and not any version. Next, under your assumptions, theorem 1 of [1] only says that for every $\epsilon>0$, the sample paths of $X$ and Hölder continuous with exponent $1-\epsilon$.

In the definition, don't you want $S_1,\ldots,s_\ell$ to be cliques?

And more generally, a necessary condition is $$f(a,b,c)f(a,b',c')f(a',b,c')f(a',b',c)=f(a',b,c)f(a,b',c)f(a,b,c')f(a',b',c')$$ for every real numbers $a,b,c,a',b',c'$.

This argument should be discretized (replacing the derivatives with rates of variations) to work with non differentiable functions.