The 1D Fibonacci quasicrystal is generated by the substitution a→ab, b→a. Its energy spectrum is a Cantor set — zero Lebesgue measure, infinitely fragmented, exhibiting self-similar gap structure.
Top: eigenvalues sorted by energy (each dot = one level). The characteristic Cantor-set fragmentation — hierarchical gaps reflecting the golden ratio — is visible even at moderate N. Bottom: density of states histogram. The self-similar gap structure persists at all scales, a hallmark of quasiperiodic order.