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.

中文翻译：

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一维替代平铺中的超均匀性和反超均匀性


这项工作考虑了表征一大类一维替代平铺中长波长涨落的超均匀性（或反超均匀性）的缩放特性。提出了一个简单的论点，它预测了控制小波数下傅里叶强度缩放的指数 α，α > 0 的平铺是超均匀的，并且数值计算证实了该预测对于准周期平铺、具有奇异连续光谱和极限的平铺是准确的。定期铺瓷砖。准周期或奇异连续情况可以用任意接近 -1 到 3 之间任何给定值的 α 来构造。极限周期平铺可以用 -1 到 1 之间的 α 或用比任何幂律更快接近零的傅里叶强度来构造。