Hidden Singularity and Fractal Dimensions for Extended Harper Operators

We prove that, for every irrational frequency, the spectra of the extended Harper family throughout the zero-measure regime have finite $1/2$-dimensional Hausdorff measure and lower box dimension at most $1/2$. Both numbers improve to $1/3$ on the isotropic critical boundary. The proof is based on a family of unitary transformations, that specializes to the chiral transform of [BJK] for the case of the critical almost Mathieu. This family allows to make every operator in this regime spectrally equivalent to a singular Jacobi operator. Using these singular representatives, we apply the method of [BJK] to construct covers by $O(q_n)$ intervals of total length $O(q_n^{-1})$, improving to $O(q_n^{-2})$ on the isotropic boundary, where $q_n$ is the denominator of a canonical approximant to $α$. The covers give upper box bounds and, with packing estimates, examples where the box-counting dimension fails to exist.