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
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