6. Fibonacci lattices have minimal dispersion on the two-dimensional torus
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Abstract
We study the size of the largest rectangle containing no point of a given point set in the two-dimensional torus, the dispersion of the point set. A known lower bound for the dispersion of any point set of cardinality n ≥ 2 in this setting is 2/n. We show that if n is a Fibonacci number then the Fibonacci lattice has dispersion exactly 2/n meeting the lower bound. Moreover, we completely characterize integration lattices achieving the lower bound and provide insight into the structure of other optimal sets. We also treat related results in the nonperiodic setting.
Abstract
We study the size of the largest rectangle containing no point of a given point set in the two-dimensional torus, the dispersion of the point set. A known lower bound for the dispersion of any point set of cardinality n ≥ 2 in this setting is 2/n. We show that if n is a Fibonacci number then the Fibonacci lattice has dispersion exactly 2/n meeting the lower bound. Moreover, we completely characterize integration lattices achieving the lower bound and provide insight into the structure of other optimal sets. We also treat related results in the nonperiodic setting.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents IX
- 1. On some recent developments in uniform distribution and discrepancy theory 1
- 2. Results and problems old and new in discrepancy theory 21
- 3. On negatively dependent sampling schemes, variance reduction, and probabilistic upper discrepancy bounds 43
- 4. Recent advances in higher order quasi-Monte Carlo methods 69
- 5. On the asymptotic behavior of the sine productΠnr =1 /2 sin πrα/ 103
- 6. Fibonacci lattices have minimal dispersion on the two-dimensional torus 117
- 7. On pair correlation of sequences 133
- 8. Some of Jiří Matoušek’s contributions to combinatorial discrepancy theory 147
- 9. Fourier analytic techniques for lattice point discrepancy 173
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents IX
- 1. On some recent developments in uniform distribution and discrepancy theory 1
- 2. Results and problems old and new in discrepancy theory 21
- 3. On negatively dependent sampling schemes, variance reduction, and probabilistic upper discrepancy bounds 43
- 4. Recent advances in higher order quasi-Monte Carlo methods 69
- 5. On the asymptotic behavior of the sine productΠnr =1 /2 sin πrα/ 103
- 6. Fibonacci lattices have minimal dispersion on the two-dimensional torus 117
- 7. On pair correlation of sequences 133
- 8. Some of Jiří Matoušek’s contributions to combinatorial discrepancy theory 147
- 9. Fourier analytic techniques for lattice point discrepancy 173
