{
  "abstract": "We present an example that shows that the double Hilbert transform $\\H$ can have the characteristic function of an open set of finite measure in $\\mathbb{R} \\times \\mathbb{R}$ as an approximate eigenvector for the eigenvalue $1$. The same holds for the dyadic model of $\\H$, the double dyadic shift. We further construct non-diagonal sets whose indicators are fixed by $\\H$ -- strips bent along the Boole curve $x=y-1/y$ and, more generally, along level curves of real Herglotz functions with purely singular measures -- and study the class of invariant sets which they generate. At positive density we construct an example built on three families of parallel lines, related to the carry cocycle $\\{u\\}+\\{v\\}-\\{u+v\\}$, and the fans -- alternate sectors of an odd number of concurrent lines -- which are the simplest members of the invariant cones, a class that we describe completely. We then study the classification of the invariant sets whose boundary lies on countably many lines. Its combinatorial half is settled completely: a locally finite arrangement of lines in which no point lies on exactly two lines is a family of parallel lines, a pencil of concurrent lines, or an affine image of the triangular lattice. The classification is further connected, via projective duality, with configurations of Sylvester--Gallai type, and, via a change of variables, with two-valued superpositions of traveling waves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15155",
  "author_total": 4,
  "authors": [
    {
      "name": "Evgeny Abakumov"
    },
    {
      "name": "Komla Domelevo"
    },
    {
      "name": "Stefanie Petermichl"
    },
    {
      "name": "Alexei Poltoratski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/57xx32b7x07j9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15155",
  "fulltext_url": null,
  "id": "57xx32b7x07j9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15155",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7213368155",
  "pdf_url": "https://arxiv.org/pdf/2609.15155",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/57xx32b7x07j9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15155",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15155"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Invariant sets of the double Hilbert transform",
  "title_text": "Invariant sets of the double Hilbert transform",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/57xx32b7x07j9",
  "venue": "arXiv",
  "versions": []
}