Low-discrepancy quasi-uniform digital nets in every dimension: blueprint
1
Introduction
1
Existence
▼
2
Graph forms and the one-digit relaxation
▶
2.1
Graph forms
2.2
The one-digit relaxation
2.3
The lattice picture
2.4
Notation
3
Frames: \(t=0\) for free
▶
3.1
The Möbius frame
3.2
The frame of \(s\) places
3.3
Designs that work for every \(k\)
3.4
Reachable designs
3.5
Cauchy identities
4
Cubic cells, \(m\equiv 0\pmod s\): designs from short lattice vectors
▶
4.1
Diagonal dominance
4.2
Admissible design columns form a lattice
4.3
The design \(U^*_s\) and the proof of Theorem A
5
Lattice skeletons: \(m=sk+r\)
▶
5.1
Fine cells and the separation reduction
5.2
Necessary conditions, forced faults and completion
5.3
Lattices
5.4
One skeleton coordinate
5.5
Two skeleton coordinates
5.6
Three or more skeleton coordinates
6
Dimension three, every resolution
▶
6.1
\(m\equiv 0\): the scramble
6.2
The net property
6.3
Cells and the geometric digit
6.4
The separation bound
6.5
Resolution \(m=3\): column nets
6.6
\(m\equiv 1,2\): lattice skeletons
6.7
Every resolution
7
Bases two and three: the deep-shift lift
8
The question of Dick, Goda and Suzuki
▶
8.1
Digital shifts
8.2
Well-separated families
8.3
The maximum norm
2
Separation constants
▶
9
The first-order deficit: certificates and ceilings
▶
9.1
The certified deficit of a design
9.2
The first-order ceiling
9.3
Dimension four
9.4
The case \(k=1\)
10
Why the deficit grows with the dimension
▶
10.1
Lower bounds for the deficit of frame designs
11
Drift and jitter
▶
11.1
Excess, drift and jitter
11.2
The holonomy, and the optimum among column nets
12
One point per cube: cube tilings and the value 1
13
Beyond the cubic value: Euclidean skeletons and half-integer layers
▶
13.1
Euclidean skeletons
13.2
Resolution \(m=3\): a layer of two cosets
13.3
Resolution \(m=3\): a rank-one layer
13.4
No half-integer layer in the frame orbit
13.5
The split frame
13.6
Separation with carries
13.7
One resolution, and searched designs
14
Open problems
A
Deferred proofs and side results
▶
A.1
A conjectural limit for one skeleton coordinate (heuristic)
A.2
Size of the constants in Theorem
37
(heuristic)
B
Per-prime designs in dimension three
Dependency graph
1 Existence
2
Graph forms and the one-digit relaxation
2.1
Graph forms
2.2
The one-digit relaxation
2.3
The lattice picture
2.4
Notation
3
Frames: \(t=0\) for free
3.1
The Möbius frame
3.2
The frame of \(s\) places
3.3
Designs that work for every \(k\)
3.4
Reachable designs
3.5
Cauchy identities
4
Cubic cells, \(m\equiv 0\pmod s\): designs from short lattice vectors
4.1
Diagonal dominance
4.2
Admissible design columns form a lattice
4.3
The design \(U^*_s\) and the proof of Theorem A
5
Lattice skeletons: \(m=sk+r\)
5.1
Fine cells and the separation reduction
5.2
Necessary conditions, forced faults and completion
5.3
Lattices
5.4
One skeleton coordinate
5.5
Two skeleton coordinates
5.6
Three or more skeleton coordinates
6
Dimension three, every resolution
6.1
\(m\equiv 0\): the scramble
6.2
The net property
6.3
Cells and the geometric digit
6.4
The separation bound
6.5
Resolution \(m=3\): column nets
6.6
\(m\equiv 1,2\): lattice skeletons
6.7
Every resolution
7
Bases two and three: the deep-shift lift
8
The question of Dick, Goda and Suzuki
8.1
Digital shifts
8.2
Well-separated families
8.3
The maximum norm