Fractal dimensions for the critical almost Mathieu operator at specific frequencies

Determine the fractal dimensions of the spectrum Sp_+(α, 1) of the critical almost Mathieu operator for α = (√5−1)/2 and for α equal to Cahen’s constant C; specifically, show that dim_H(Sp_+((√5−1)/2, 1)) = dim_B(Sp_+((√5−1)/2, 1)) = 1/2, and that dim_H(Sp_+(C, 1)) < 1/2 = \underline{dim}_B(Sp_+(C, 1)) < 2/3 = \overline{dim}_B(Sp_+(C, 1)).

Background

The critical almost Mathieu operator has been extensively studied, with the spectrum known to be a Cantor set of zero Lebesgue measure for irrational frequencies; however, precise fractal dimensions are largely unresolved beyond certain bounds and special cases.

The authors’ computations suggest the golden-ratio case might have fractal dimension one-half, while the Cahen constant may exhibit strict inequality between Hausdorff and box-counting dimensions, indicating nontrivial multifractal behavior.