Regularities of the distribution of abstract van der Corput sequences

TL;DR

This paper studies the distribution properties of abstract van der Corput sequences, providing explicit discrepancy calculations and characterizing bounded remainder sets, thus advancing understanding of their uniformity in abstract numeration systems.

Contribution

It offers explicit discrepancy formulas and characterizes bounded remainder sets for abstract van der Corput sequences, extending classical results to a more general setting.

Findings

Explicit discrepancy function computed

Bounded remainder sets characterized

Sequences shown to be low discrepancy under certain conditions

Abstract

Similarly to $\beta$-adic van der Corput sequences, abstract van der Corput sequences can be defined for abstract numeration systems. Under some assumptions, these sequences are low discrepancy sequences. The discrepancy function is computed explicitely, and a characterization of bounded remainder sets of the form $[0,y)$ is provided.