Improved dispersion bounds for modified Fibonacci lattices☆
Section snippets
Introduction and main results
We consider point sets in the unit square consisting of (not necessarily distinct) elements. We are interested in the size of the largest box amidst such point sets which does not contain any points of . We speak of this size as the (standard) dispersion of the point set . More formally, we introduce the set of all axes-parallel boxes in the unit square; i.e. . The dispersion of can then be defined as , where
Auxiliary results
We begin by recalling some identities involving the golden ratio. First is . For we have the beautiful relation between Fibonacci numbers and powers of the golden ratio, which can be easily shown by induction on . We will use this relation several times throughout this paper. We always assume in the following.
Lemma 6 For as introduced in Definition 1 we have
Proof The essential observation for the proof of this lemma is the following: For define the disjoint
Proof of Theorem 4
First we adapt the proof in [4] to get the following proposition:
Proposition 9 The maximal periodic boxes amidst the points of are of the form for and certain integers and where the indices are taken modulo .
Proof This has essentially been proven in [4] but has not been stated there explicitly. We show how to derive this result from [4, Lemma 6.8]. We consider the function from Definition 1. Obviously is -periodic; i.e. for every . For integers
Improvements in dispersion and discrepancy
We can improve the dispersion of even further by altering the values for and . It seems reasonable to choose those gaps such that the size of the largest maximal exterior boxes matches the area of the interior boxes. Let for some positive value we specify below. We set and for as in Definition 1 and define with respect to these choices of and . For small we found numerically that must be chosen as follows:
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If is odd,
Conclusion and unanswered questions
Our results show that Until now the right limit of this interval was 2. The exact value of remains unknown, but we conjecture that is optimal. We believe that, given the success of the Fibonacci lattice with respect to other measures of uniformity, especially the torus dispersion, it is reasonable to assume that an optimal point set with respect to dispersion would be similar to such lattices. Moreover, the property that every maximal
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2021, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
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Communicated by A. Hinrichs.