We present mathematical theory for self-similarity induced spectral gaps in the spectra of systems generated by generalised Fibonacci tilings. Our results characterise super band gaps, which are spectral gaps that exist for all sufficiently large periodic systems in a Fibonacci-generated sequence. We characterise super band gaps in terms of a growth condition on the traces of the associated transfer matrices. Our theory includes a large family of generalised Fibonacci tilings, including both precious mean and metal mean patterns. We apply our analytic results to characterise spectra in three different settings: compressional waves in a discrete mass-spring system, axial waves in structured rods and flexural waves in multi-supported beams. The theory is shown to give accurate predictions of the super band gaps, with minimal computational cost and significantly greater precision than previous estimates. It also provides a mathematical foundation for using periodic approximants (supercells) to predict the transmission gaps of quasicrystalline samples, as we verify numerically.

我们提出了由广义斐波那契平铺生成的系统频谱中自相似引起的频谱间隙的数学理论。我们的结果描述了超级带隙的特征，超级带隙是斐波那契生成序列中所有足够大的周期系统都存在的谱隙。我们根据相关转移矩阵迹线的生长条件来表征超带隙。我们的理论包括一大类广义斐波那契排列，包括珍贵均值和金属均值模式。我们应用分析结果来表征三种不同设置下的光谱：离散质量弹簧系统中的压缩波、结构杆中的轴向波和多支撑梁中的弯曲波。该理论被证明能够以最小的计算成本准确预测超级带隙，并且比之前的估计精度显着提高。正如我们通过数值验证的那样，它还为使用周期性近似（超晶胞）来预测准晶样品的传输间隙提供了数学基础。