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 $$ \limsup_{n\to\infty} \frac{1}{\log n} S_n(\theta,\phi) = c\ ? $$

  • $\begingroup$ This can only hold if $\int_{-\pi}^{\pi} \ln |e^{i\varphi}-1|\, d\varphi = 0$ (by the ergodic theorem), so I guess this is clear?! $\endgroup$ Sep 6, 2014 at 18:58
  • $\begingroup$ On second thoughts, this is indeed clear since the integrand is harmonic. $\endgroup$ Sep 6, 2014 at 19:01
  • $\begingroup$ Did you ever resolve this question, Doug? $\endgroup$ Oct 29, 2016 at 2:01
  • $\begingroup$ No, although the numerical evidence was pretty convincing, at least for some "random" values of $\theta$ and $\phi$. Are you asking for any particular reason? $\endgroup$ Nov 3, 2016 at 13:30


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