Liminf-results for sums with Kronecker sequence

For irrational $\theta$ and 1-periodic function $f$ we consider sums ${\sum }_0^{Q-1}f(k\theta +\phi )$ where $\phi \in \mathbb{R}$. Sidorov proved that if $f$ is absolutely continuous function, then ${\mathrm{lim\hspace{0.17em}inf}}_{Q\to \mathrm{\infty }}\mathrm{\mid }{\sum }_0^{Q-1}f(k\theta +\phi )\mathrm{\mid }=0$ for any irrational $\theta$ and any $\phi \in \mathbb{R}$. The article shows that this property is not a criterion of absolute continuity, and also obtains some other results concerning the liminf-properties of these sums.