On the one-sided boundedness of the local discrepancy of $\{n\alpha \}$-sequences
Volume 206 / 2022
Acta Arithmetica 206 (2022), 97-114
MSC: Primary 11K38; Secondary 11J70.
DOI: 10.4064/aa211015-12-11
Published online: 5 December 2022
Abstract
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 \[D_n(\alpha ,c)=\sum _{j=1}^n 1_{\{\{j\alpha \} \lt c\}}-cn.\] 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-sidedly bounded are derived. Using these, certain topological properties are given to describe the size of the set \[O_c=\{\alpha \in \mathbb R^+\setminus \mathbb Q: (D_n(\alpha ,c)) \text { is one-sidedly bounded}\}.\]
