2025/12/22 by Rodolfo Viera, Viera, Rodolfo
Materials Science · Mathematics · #Quasicrystal Structures and Properties #Mathematical Approximation and Integration #Analytic and geometric function theory
paper · doi:10.48550/arxiv.2512.19854
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 ℝd is uniformly diffractive, converges rapidly enough to a discrete set X in ℝd, and their diffraction measures \widehatγXn are asymptotically orthogonal with respect to the Lebesgue measure in ℝd, 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 ℝd. 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.