Density fluctuations in many-body systems are of fundamental importance throughout various scientic disciplines, including physics, materials science, number theory and biology. Hyperuniform systems, which include crystals and quasicrystals, have density fluctuations that are anomalously suppressed at long wavelengths compared to the fluctuations in typical disordered point distributions such as in ideal gases and liquids. Quantitatively speaking, hypperuniform systems are characterized by a local number variance of points within a spherical window of radius R that grows more slowly than the window volume in the large-R limit.

In this talk, we provide the first rigorous hyperuniformity analyses of quasicrystals by employing a new criterion for hyperuniformity to quantitatively characterize quasicrystalline

point sets. We reveal that one-dimensional quasicrystals produced by projection from a two-dimensional lattice fall into two distinct classes with respect to their large-scale density fluctuations. Depending on the width of the projection window, the number variance is either uniformly bounded in the one class for large R, or it scales like ln R in the other class. This distinction provides a new classication of one-dimensional quasicrystalline systems and suggests that measures of hyperuniformity may define new classes of quasicrystals in higher dimensions as well.