A van der Corput-type algorithm for LS-sequences of points

Abstract

In this paper we associate to any $LS$-sequence of partitions ${\rho_{L,S}^n}$ the corresponding $LS$-sequence of points ${\xi_{L,S}^n}$ obtained reordering the points of each partition with an explicit algorithm. The procedure begins with the representation in base $L+S$ of natural numbers, $[n]_{L+S}$, and ends with the $LS$-radical inverse function $\phi_{L,S}$, introduced ad hoc, evaluated at an appropriate subsequence of natural numbers depending on $L$ and $S$. This construction is deeply related to the geometric representation of the points of ${\xi_{L,S}^n}$ by suitable affine functions and reminds the van der Corput sequences in base $b$. Keywords: Uniform distribution, sequences of partitions, van der Corput sequences, discrepancy.