On the one-sided boundedness of the local discrepancy of $\{n\alpha \}$-sequences

The main interest of this article is the one-sided boundedness of the local discrepancy of $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ on the interval $(0,c)\subset (0,1)$ defined by $$D_n(\alpha ,c)=\sum _{j=1}^n{1}_{\{\{j\alpha \}<c\}}-cn\mathrm{.}$$ We focus on the special case $c\in (0,1)\cap \mathbb{Q}$. Several necessary and sufficient conditions on $\alpha$ for $(D_n(\alpha ,c))$ to be one-side bounded are derived. Using these, certain topological properties are given to describe the size of the set $$O_c=\{\alpha \in \text{irr}:(D_n(\alpha ,c))\text{ is one-side bounded}\}\mathrm{.}$$