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Fractal-dimension equality and min-formula for d-dimensional Fibonacci Hamiltonians

Prove that, for every dimension d ≥ 1 and coupling λ > 0, the Hausdorff dimension and the box-counting dimension of the spectrum of the d-dimensional Fibonacci Hamiltonian H_λ^{(d)} are equal and satisfy dim_H(Sp(H_λ^{(d)})) = dim_B(Sp(H_λ^{(d)})) = min{1, d · dim_B(Sp(H_λ))}.

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Background

Building on extensive numerical computations and known one-dimensional theory (e.g., dynamical Cantor sets and dimension properties for H_λ), the authors conjecture a precise characterization of the spectral fractal dimensions in higher dimensions.

This conjecture extends beyond the previously known d = 2 case for almost all λ and asserts equality of Hausdorff and box-counting dimensions along with the min{1, d * dim_B(Sp(H_λ))} formula, offering a unified picture of spectral fractal behavior for multi-dimensional Fibonacci quasicrystals.

References

We conjecture that the box-counting and Hausdorff dimensions of the spectrum of the $d$-dimensional Fibonacci Hamiltonian are equal and satisfy \begin{equation} \label{eq:dDimFibConj} \mathrm{dim}{\mathrm{H}(Sp(H\lambda{(d)}))=\mathrm{dim}{\mathrm{B}(Sp(H\lambda{(d)}))=\min\left{1,d\cdot\mathrm{dim}{\mathrm{B}(H{\lambda})\right}. \end{equation}

— Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals (2407.20353 - Colbrook et al., 29 Jul 2024) in Section 7: Final Remarks (Conjectures), equation (dDimFibConj)