Stability and robustness of mathematical quasicrystals under statistical convergence

Abstract

In this work we address the stability and robustness of uniformly discrete point sets in Euclidean spaces. Firstly, we prove that if a sequence of point configurations contained in $\mathbb{R}^d$ is uniformly diffractive, converges rapidly enough to a discrete set $X$ in $\mathbb{R}^d$, and their diffraction measures $\widehat{γ_{X_n}}$ are asymptotically orthogonal with respect to the Lebesgue measure in $\mathbb{R}^d$, then $X$ is necessarily a quasicrystal. The convergence is addressed for a distance that quantifies the statistical closeness between two uniformly discrete point sets in $\mathbb{R}^d$. Secondly, motivated by their applications in the diffraction theory of quasicrystals, we establish the continuity of the Fourier Transform of quasicrystals in this topology. This continuity result, in turn, allows us to rigorously demonstrate that well-known robustness properties of quasicrystals against random errors remain stable under the statistical convergence considered. Some applications for rapidly-solidified quasicrystals are highlighted.