Uniform distribution of sequences of points and partitions

TL;DR

This paper develops explicit methods for constructing uniformly distributed sequences of points and partitions on fractals and analyzes their discrepancy, with applications in Quasi-Monte Carlo methods.

Contribution

It introduces algorithms for creating u.d. sequences on fractals using IFS and $ ho$-refinements, and provides discrepancy bounds for these sequences.

Findings

Constructed u.d. sequences on fractals satisfying OSC.

Provided discrepancy estimates for generalized Kakutani's sequences.

Extended discrepancy analysis to a broad class of fractals.

Abstract

The interest for uniformly distributed (u.d.) sequences of points, in particular for sequences with small discrepancy, arises from various applications. For instance, low-discrepancy sequences, which are sequences with a discrepancy of order $((\log N)^d)/N$ ($d$ is the dimension of the space where the sequence lies), are a fundamental tool for getting faster rate of convergence in approximation involving Quasi-Monte Carlo methods. The objectives of this work can be summarized as follows (1)The research of explicit techniques for introducing new classes of u.d. sequences of points and of partitions on $[0,1]$ and also on fractal sets (2) A quantitative analysis of the distribution behaviour of a class of generalized Kakutani's sequences on $[0,1]$ through the study of their discrepancy. Concerning (1), we propose an algorithm to construct u.d. sequences of partitions and of points on fractals generated by an Iterated Function System (IFS) of similarities having the same ratio and satisfying a natural separation condition of their components called Open Set Condition (OSC). We also provide an estimate for the elementary discrepancy of these sequences. We generalize these results to a wider class of fractals by using a recent generalization of Kakutani's splitting procedure on $[0,1]$, namely the technique of $\rho-$refinements. First, we focus on (2) and get precise bounds for the discrepancy of a large class of generalized Kakutani's sequences, exploiting a correspondence between the tree representation associated to successive $\rho-$refinements and the tree generated by Khodak's coding algorithm. Then we adapt the $\rho-$refinements method to the new class of fractals and prove bounds for the elementary discrepancy of the sequences of partitions constructed with such a procedure.