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Abstract
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The aim of this paper is to develop greedy algorithms which generate uniformly distributed sequences
in the
-dimensional
unit cube
.
The figures of merit are three different variants of the
discrepancy. Theoretical results along with numerical experiments suggest that the
resulting sequences have excellent distribution properties. The approach we follow
here is motivated by recent work of Steinerberger and Pausinger who consider similar
greedy algorithms, where they minimize functionals that can be related to the star
discrepancy or energy of point sets. In contrast to many greedy algorithms, where the
resulting elements of the sequence can only be given numerically, we will find that in the
one-dimensional case our algorithms yield rational numbers which we can describe
precisely. In particular, we will observe that any initial segment of a sequence in
can be naturally
extended to a uniformly distributed sequence where all subsequent elements are of the form
for some
. We will also investigate
the dependence of the
discrepancy of the resulting sequences on the dimension
.
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Keywords
uniform distribution modulo 1, $L_2$ discrepancy, diaphony,
van der Corput sequence, greedy algorithm
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Mathematical Subject Classification
Primary: 11K06, 11K31, 11K38
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Milestones
Received: 24 October 2021
Accepted: 7 January 2022
Published: 15 October 2022
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