Substitutions and 1/2-Discrepancy of $\{n \theta + x\}$ \rm{II}

TL;DR

This paper investigates the ergodic properties of a renormalization process related to the 1/2-discrepancy sums of rotations, providing insights into their growth behavior and bounds.

Contribution

It introduces a renormalization approach to analyze the discrepancy sums of rotations, revealing new ergodic properties and growth rate bounds.

Findings

Established ergodic properties of the renormalization procedure

Derived almost-sure bounds on discrepancy sum growth rates

Linked discrepancy behavior to rotation dynamics

Abstract

Ergodic properties of a renormalization procedure for studying the 1/2-discrepancy sums driven by rotations are studied, with corresponding implications for almost-sure bounds on the growth rates for these discrepancy sums.