Let $\theta, \phi \in [0,1)$, and consider the sums $$ S_n(\theta,\phi)=\sum_{k=0}^n \log|e^{2\pi i (k\theta+\phi)}-1|. $$ The possible boundedness from above of such sums plays a key role in analyzing the expansiveness of certain algebraic actions of the discrete Heisenberg group, which is why I'm interested. However, some numerical experiments indicate that for typical $\theta$ and $\phi$ these sums occasionally get as large as a constant times $\log n$.

Question: Is it true that for almost every pair $\theta,\phi$ there is a $c=c(\theta,\phi)$ with $0<c<\infty$ such that $$ \limsup_{n\to\infty} \frac{1}{\log n} S_n(\theta,\phi) = c\ ? $$