Hyperuniformity and anti-hyperuniformity in one-dimensional substitution
We consider the scaling properties characterizing the hyperuniformity (or
anti-hyperuniformity) of long wavelength fluctuations in a broad class of
one-dimensional substitution tilings. We present a simple argument that
predicts the exponent $\alpha$ governing the scaling of Fourier intensities at
small wavenumbers, tilings with $\alpha>0$ being hyperuniform, and confirm with
numerical computations that the predictions are accurate for quasiperiodic
tilings, tilings with singular continuous spectra, and limit-periodic tilings.
Tilings with quasiperiodic or singular continuous spectra 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.