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Hyperuniform point sets on projective spaces

Johann S. Brauchart, Peter J. Grabner

Abstract

We extend the notion of hyperuniformity to the projective spaces $\mathbb{RP}^{d-1}$, $\mathbb{CP}^{d-1}$, $\mathbb{HP}^{d-1}$, and $\mathbb{OP}^2$. We show that hyperuniformity implies uniform distribution and present examples of deterministic point sets as well as point processes which exhibit hyperuniform behaviour.

Hyperuniform point sets on projective spaces

Abstract

We extend the notion of hyperuniformity to the projective spaces , , , and . We show that hyperuniformity implies uniform distribution and present examples of deterministic point sets as well as point processes which exhibit hyperuniform behaviour.
Paper Structure (12 sections, 6 theorems, 67 equations, 1 table)

This paper contains 12 sections, 6 theorems, 67 equations, 1 table.

Table of Contents

  1. Introduction
  2. Hyperuniformity on compact doubly homogeneous spaces
  3. Harmonic analysis on compact doubly homogeneous spaces
  4. Hyperuniformity on $\mathbb{FP}^{d-1}$
  5. Uniform distribution
  6. Hyperuniformity for large balls implies uniform distribution
  7. Hyperuniformity for small balls implies uniform distribution
  8. Hyperuniformity for balls at threshold order implies uniform distribution
  9. Examples
  10. Maximisers of the sum of distances
  11. Jittered sampling
  12. The harmonic ensemble

Key Result

Theorem 1

Let $(X_N)_{N\in\mathbb{N}}$ be hyperuniform for large balls. Then $(X_N)_{N\in\mathbb{N}}$ is uniformly distributed. More precisely, holds for all $n\geq1$.

Theorems & Definitions (14)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Remark 4
  • Theorem 3
  • ...and 4 more