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Stability of mathematical quasicrystals under statistical convergence

Rodolfo Viera

TL;DR

This work addresses the stability of mathematical quasicrystals under statistical convergence. It introduces a statistical Gromov-Hausdorff distance $\\rho_{\\mathsf{stat}}$ on uniformly discrete sets and proves that a uniformly separated, uniformly quasicrystalline sequence $X_n$ converging rapidly to a limit $X$ with the same separation radius yields a limit that is a quasicrystal; it also proves the continuity of the Fourier transform along such convergent families, enabling Fourier-recoverability under random perturbations. The results establish that both the diffractive nature and the spectral type of quasicrystals are preserved under this notion of convergence, and that robustness against random perturbations persists in the limit. Collectively, the paper provides rigorous stability guarantees for quasicrystal models under deterministic and stochastic perturbations, with implications for diffraction analysis and the study of aperiodic order.

Abstract

In this work, we prove that if a uniformly separated sequence in $\mathbb{R}^d$ is uniformly quasicrystalline and converges rapidly enough to a discrete set $X$ in $\mathbb{R}^d$ having the same separation radius as the sequence, then $X$ is also a quasicrystal. The convergence is addressed for a distance that quantifies the statistical closeness between two uniformly discrete point sets in $\mathbb{R}^d$. Furthermore, motivated by the robustness of quasicrystals under random perturbations, we establish the continuity, for this distance, of the Fourier Transform of quasicrystals. This continuity result, in turn, allows us to rigorously demonstrate that established robustness properties of quasicrystals against random errors remain stable under the statistical convergence considered.

Stability of mathematical quasicrystals under statistical convergence

TL;DR

This work addresses the stability of mathematical quasicrystals under statistical convergence. It introduces a statistical Gromov-Hausdorff distance on uniformly discrete sets and proves that a uniformly separated, uniformly quasicrystalline sequence converging rapidly to a limit with the same separation radius yields a limit that is a quasicrystal; it also proves the continuity of the Fourier transform along such convergent families, enabling Fourier-recoverability under random perturbations. The results establish that both the diffractive nature and the spectral type of quasicrystals are preserved under this notion of convergence, and that robustness against random perturbations persists in the limit. Collectively, the paper provides rigorous stability guarantees for quasicrystal models under deterministic and stochastic perturbations, with implications for diffraction analysis and the study of aperiodic order.

Abstract

In this work, we prove that if a uniformly separated sequence in is uniformly quasicrystalline and converges rapidly enough to a discrete set in having the same separation radius as the sequence, then is also a quasicrystal. The convergence is addressed for a distance that quantifies the statistical closeness between two uniformly discrete point sets in . Furthermore, motivated by the robustness of quasicrystals under random perturbations, we establish the continuity, for this distance, of the Fourier Transform of quasicrystals. This continuity result, in turn, allows us to rigorously demonstrate that established robustness properties of quasicrystals against random errors remain stable under the statistical convergence considered.
Paper Structure (7 sections, 14 theorems, 100 equations)

This paper contains 7 sections, 14 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Measures and Diffraction Theory
  3. Statistical closeness and convergence
  4. Proofs of Theorems \ref{['thm: stabquasi']} and \ref{['thm: stabrecov']}
  5. Stability of the diffractive nature of discrete point sets
  6. On spectral stability under convergence
  7. Stability of robustness under random perturbations

Key Result

Theorem A

Let $r_0>0$ and $d\in\mathbb N$ be fixed. Let $X\in\mathcal{UD}_{r_0,d}$ and $(X_n)_{n\geq 1}\subset\mathcal{UD}_{r_0,d}$ be a uniformly quasicrystalline sequence in $\mathbb R^d$ such that Then $X$ is a quasicrystal.

Theorems & Definitions (30)

  • Theorem A
  • Theorem B
  • Proposition 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • ...and 20 more