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Validity of box-counting dimension formula for d-dimensional Fibonacci Hamiltonians

Determine whether, for every dimension d ≥ 1 and coupling λ > 0, the box-counting dimension of the spectrum of the d-dimensional Fibonacci Hamiltonian H_λ^{(d)} equals min{1, d · dim_B(Sp(H_λ))}.

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Background

The authors compute fractal dimensions for one-, two-, and three-dimensional Fibonacci Hamiltonians using periodic covers under assumption (S1) and present numerical evidence that the box-counting dimension scales like d/log(λ) for large λ. This evidence suggests the formula dim_B(Sp(H_λ{(d)})) = min{1, d * dim_B(Sp(H_λ))}.

While this equality is known for d = 2 for all but countably many λ (based on Yessen’s work), its general validity for arbitrary d remains undetermined. Establishing it would clarify how spectral complexity scales with dimension in quasicrystal models.

References

It is not known whether this formula holds, apart from the case of $d=2$, where it is known to hold for all but countably many $\lambda$.

— Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals (2407.20353 - Colbrook et al., 29 Jul 2024) in Section: The Fibonacci Hamiltonian (sec:Fibonacci_numerics), after Figure FIBfig1