Liminf-results for sums with Kronecker sequence

For irrational $\theta$ and 1-periodic function $f$ we consider sums $\sum_0^{Q-1}f(k\theta+\varphi)$ where $\varphi \in \mathbb R$. Sidorov proved that if $f$ is absolutely continuous function, then $\liminf_{Q \to \infty} |\sum_0^{Q-1}f(k\theta+\varphi)| = 0$ for any irrational $\theta$ and any $\varphi \in \mathbb R$. 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.