$LS$-successioni di punti nel quadrato

TL;DR

This thesis explores generalizations of $LS$-sequences of points in the unit square, including new algorithms and their graphical representations, with a focus on uniform distribution and discrepancy theory.

Contribution

It introduces two new generalizations of $LS$-sequences in the unit square and implements an algorithm for graphical visualization.

Findings

Two new $LS$-sequence generalizations in the unit square.

An implemented algorithm for sequence visualization.

Analysis of uniform distribution properties.

Abstract

The main purpose of this master thesis is to study the $LS$-sequences of points introduced by Carbone in \cite{Carbone} and find two generalizations of them to the unit square. Here we also present a new algorithm proposed by the same author in \cite{Carbone2} and we implement it in order to have a graphical description of these sequences. Chapter 1 includes a collection of results concerning the uniform ditribution theory and the discrepancy (we refer to \cite{Drmota_Tichy} and \cite{Kuipers_Niederreiter} for a complete survey on the matter). In Chapter 2 we focuse our attention on the $LS$-sequences of partitions and of points in the unit interval, giving particular attention to the ordering of the points "\`{a} la van der Corput \rq\rq and finding a way to compute them related to the digit expansion of natural numbers in base $L+S$. In Chapter 3 we move on the unit square where we find two generalizations of the $LS$-sequences following the historical development of the van der Corput sequence in the multidimensional case. This is the reason why we call the first sequence $LS$- sequence of points \`{a} la van der Corput-Hammersley and the second one $LS$-sequence \`{a} la Halton.