Discrepancy of generalized $LS$-sequences

TL;DR

This paper analyzes the discrepancy bounds of $LS$-sequences, providing explicit constants, and introduces a new class of generalized $LS$-sequences in the unit interval.

Contribution

It offers explicit discrepancy bounds for $LS$-sequences and introduces a generalized construction of these sequences.

Findings

Explicit discrepancy bounds with constants for $LS$-sequences.

Construction of a new class of generalized $LS$-sequences.

Enhanced understanding of low-discrepancy sequence properties.

Abstract

The $LS$-sequences are a parametric family of sequences of points in the unit interval. They were introduced by Carbone, who also proved that under an appropriate choice of the parameters $L$ and $S$, such sequences are low-discrepancy. The aim of the present paper is to provide explicit constants in the bounds of the discrepancy of $LS$-sequences. Further, we generalize the construction of Carbone and construct a new class of sequences of points in the unit interval, the generalized $LS$-sequences.