Oğuz, EC; Socolar, JES; Steinhardt, PJ; Torquato, S, *Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings.*,
Acta Crystallographica. Section A, Foundations and Advances, vol. 75 no. Pt 1
(January, 2019),
pp. 3-13 [doi] .
**Abstract:**

*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.*