  We study the interplay between quasi-periodic disorder and superconductivity
in a 1D tight-binding model with the quasi-periodic modulation of on-site
energies that follow the Fibonacci rule and all the eigenstates are
multifractal. As a signature of multifractality, we observe the power-law
dependence of the correlation between different single-particle eigenstates as
a function of their energy difference. By computing numerically the
superconducting transition temperature, we find the distribution of critical
temperatures, analyze their statistics and estimate the mean value and variance
of critical temperatures for various regimes of the attractive coupling
strength and quasi-periodic disorder. We find an enhancement of the critical
temperature as compared to the analytical results that are based on strong
assumptions of absence of correlations and self-averaging of multiple
characteristics of the system, which are not justified for the Fibonacci chain.
For the very weak coupling regime, we observe a crossover where the
self-averaging of the critical temperature breaks down completely and a strong
sample-to-sample fluctuations emerge.
