{
  "abstract": "We investigate the relationship between weak asymptotic symmetry ($\\mathrm{WAS}$) and asymptotic symmetry ($\\mathrm{AS}$) for quasisymmetric maps associated with planar quasilines. To this end, we introduce a formally weaker condition, called equidistant weak asymptotic symmetry ($\\mathrm{EWAS}$), and prove that, for quasisymmetric embeddings of $\\mathbb{R}$ into $\\mathbb{C}$, the three conditions $\\mathrm{AS}$, $\\mathrm{WAS}$, and $\\mathrm{EWAS}$ are equivalent. We then establish that $\\mathrm{WAS}$ implies $\\mathrm{AS}$ for quasisymmetric homeomorphisms from an arbitrary quasiline onto $\\mathbb{R}$. More generally, if $h\\colonΓ_1\\toΓ_2$ is a quasisymmetric homeomorphism between quasilines and $Γ_1$ is asymptotically conformal, then $\\mathrm{WAS}$ implies $\\mathrm{AS}$. A formally dual statement holds when $Γ_2$ is asymptotically conformal, provided that $h$ is uniformly continuous. In the compact setting of bounded quasicircles, these results yield an answer to the $\\mathrm{WAS}$-$\\mathrm{AS}$ problem posed by Brania and Yang.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19733",
  "author_total": 2,
  "authors": [
    {
      "name": "Katsuhiko Matsuzaki"
    },
    {
      "name": "Fei Tao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fkv7b7vtk3sss.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19733",
  "fulltext_url": null,
  "id": "fkv7b7vtk3sss",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19733",
  "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.CV"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7213580039",
  "pdf_url": "https://arxiv.org/pdf/2609.19733",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fkv7b7vtk3sss.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19733",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19733"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Weak Asymptotic Symmetry of Quasisymmetric Embeddings on Quasilines",
  "title_text": "Weak Asymptotic Symmetry of Quasisymmetric Embeddings on Quasilines",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/fkv7b7vtk3sss",
  "venue": "arXiv",
  "versions": []
}