“Hyperuniformity and Anti-Hyperuniformity in One-Dimensional Substitution Tilings” is published in Acta Crystalographica Section A

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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 \(\alpha\) governing the scaling of Fourier intensities at small wavenumbers, tilings with \(\alpha>0\) 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 \(\alpha\) arbitrarily close to any given value between \(-1\) and \(3\). Limit-periodic tilings can be constructed with \(\alpha\) between \(-1\) and \(1\) or with Fourier intensities that approach zero faster than any power law.