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PreprintCC BY 4.0arXiv

Hidden Singularity and Fractal Dimensions for Extended Harper Operators

  • Simon Becker
  • Svetlana Jitomirskaya
  • Igor Krasovsky
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Abstract

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(qn) intervals of total length O(qn−1), improving to O(qn−2) on the isotropic boundary, where qn 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.

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Cite

BibTeXRIS
BibTeX
@misc{becker2026hidden,
  title = {{Hidden Singularity and Fractal Dimensions for Extended Harper Operators}},
  author = {Simon Becker and Svetlana Jitomirskaya and Igor Krasovsky},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35822},
  eprint = {2610.35822},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35822},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Hidden Singularity and Fractal Dimensions for Extended Harper Operators
AU  - Simon Becker
AU  - Svetlana Jitomirskaya
AU  - Igor Krasovsky
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35822
UR  - https://arxiv.org/abs/2610.35822
AB  - 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.
ER  -