{
  "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 $O(q_n)$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35822",
  "author_total": 3,
  "authors": [
    {
      "name": "Simon Becker"
    },
    {
      "name": "Svetlana Jitomirskaya"
    },
    {
      "name": "Igor Krasovsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2d2d9hs1mj1et.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35822",
  "fulltext_url": null,
  "id": "2d2d9hs1mj1et",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35822",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mes-hall"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7223845479",
  "pdf_url": "https://arxiv.org/pdf/2610.35822",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2d2d9hs1mj1et.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.35822",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35822"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mes-hall",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Hidden Singularity and Fractal Dimensions for Extended Harper Operators",
  "title_text": "Hidden Singularity and Fractal Dimensions for Extended Harper Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/2d2d9hs1mj1et",
  "venue": "arXiv",
  "versions": []
}