Hyperuniformity and anti‐hyperuniformity in one‐dimensional substitution tilings

This work considers the scaling properties characterizing the hyperuniformity (or anti‐hyperuniformity) of long‐wavelength fluctuations in a broad class of one‐dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit‐periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between −1 and 3. Limit‐periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law. This work examines the long‐wavelength scaling properties of self‐similar substitution tilings, placing them in their hyperuniformity classes. Quasiperiodic, non‐PV (Pisot–Vijayaraghavan number) and limit‐periodic examples are analyzed. Novel behavior is demonstrated for certain limit‐periodic cases.