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

TL;DR

This paper investigates conditions under which the local discrepancy of irrational rotations, restricted to a rational interval, remains bounded on one side, and characterizes the size of the set of such irrationals.

Contribution

It provides necessary and sufficient conditions for one-sided boundedness of local discrepancy and describes the topological size of the set of irrationals satisfying these conditions.

Findings

Characterizes when the local discrepancy is one-sided bounded for given .

Identifies topological properties of the set of irrationals with bounded discrepancy.

Provides conditions linking , , and the irrationality of .

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 \[D_n(\alpha,c)=\sum_{j=1}^n 1_{\{\{j\alpha\}<c\}}-cn.\] 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-side bounded are derived. Using these, certain topological properties are given to describe the size of the set \[O_c=\{\alpha\in \irr: (D_n(\alpha,c)) \text{ is one-side bounded}\}.\]