{
  "abstract": "We consider, for any irrational $α$ and interval $I \\subset \\mathbb{T}$, the 2-interval coding subshift $X^{(I, α)}$ induced by coding orbits under repeated rotation by $α$ via membership in $I$ or $I^c$. Each sequence $c \\in X^{(I, α)}$ has an associated Schrödinger operator $H_c$, and in \\cite{kaminaga} it was proved that if the continued fraction of $α$ has digits with limsup at least $4$, then almost every $c \\in X^{(I, α)}$ has so-called $3$-block Gordon structure, which implies that the operator $H_c$ has no eigenvalues. We significantly improve this result by completely characterizing almost-sure $3$-block Gordon structure, proving that in fact it holds for all $(α,I)$ except for a countable set of pairs $(α, |I|)$ where $α$ is Möbius equivalent to the silver mean and $|I| \\in \\mathbb{Z}α+ f(α)$ where $f(α)$ is a specific infinite series taking value either $\\frac{1}{2}, \\fracα{2}$, or $\\frac{α+1}{2}$. We also show that for a set of $α$ of full measure, and for every $I$, the set of points whose orbit codings do not have $3$-block Gordon structure has Hausdorff dimension bounded away from $1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00301",
  "author_total": 2,
  "authors": [
    {
      "name": "Luke Hetzel"
    },
    {
      "name": "Ronnie Pavlov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/00vw8fza9xgj1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00301",
  "fulltext_url": null,
  "id": "00vw8fza9xgj1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00301",
  "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",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7206202727",
  "pdf_url": "https://arxiv.org/pdf/2609.00301",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/00vw8fza9xgj1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00301",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00301"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Periodic structure and Schrodinger operators for codings of circle rotations",
  "title_text": "Periodic structure and Schrodinger operators for codings of circle rotations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/00vw8fza9xgj1",
  "venue": "arXiv",
  "versions": []
}