Stability of mathematical quasicrystals under statistical convergence

TL;DR

This paper proves that under certain conditions, quasicrystals remain stable when approximated by sequences converging statistically, and shows their Fourier Transform's continuity ensures robustness against random perturbations.

Contribution

It introduces a new statistical convergence framework for quasicrystals and proves their stability and Fourier Transform continuity under this convergence.

Findings

Quasicrystals are stable under rapid statistical convergence.

Fourier Transforms of quasicrystals are continuous with respect to the statistical distance.

Robustness of quasicrystals against random errors is preserved under this convergence.

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.