{
  "abstract": "Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold without boundary, and let $\\{e_j\\}_{j=0}^\\infty$ be an orthonormal basis of Laplace eigenfunctions with frequencies $\\{λ_j\\}_{j=0}^\\infty$. For a domain $Ω\\subset M$, consider $$ N_Ω(λ) = \\sum_{λ_j\\leqλ} \\left|\\int_Ωe_j\\,dV_g\\right|^2. $$ Suppose that $\\partialΩ$ is $s$-Ahlfors regular, with $s\\in{[d-1,d)}$, and that $Ω$ satisfies a two-sided corkscrew condition. We prove that $$ N_Ω(λ) = \\operatorname{vol}(Ω)+O\\bigl(λ^{s-d}\\bigr), $$ and that this remainder is sharp. We further identify the geometric quantity governing the second term: an exact asymptotic of order $λ^{s-d}$ holds precisely when the boundary admits a natural $s$-dimensional crossing volume $\\mathcal V^s(\\partialΩ)$, in which case $$ N_Ω(λ) = \\operatorname{vol}(Ω) - C_{d,s}\\, \\mathcal V^s(\\partialΩ)\\, λ^{s-d} + o\\bigl(λ^{s-d}\\bigr). $$ For admissible self-similar boundaries, the crossing volume exists in the non-lattice case, and hence the above two-term asymptotic holds. In the lattice case, the second-order behavior is instead governed by a log-periodic profile.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08487",
  "author_total": 2,
  "authors": [
    {
      "name": "Xucheng Hu"
    },
    {
      "name": "Yakun Xi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/afq7g2q1k1636.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08487",
  "fulltext_url": null,
  "id": "afq7g2q1k1636",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08487",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7221008047",
  "pdf_url": "https://arxiv.org/pdf/2610.08487",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/afq7g2q1k1636.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08487",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08487"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Kuznecov formulas for domains with fractal boundary",
  "title_text": "Kuznecov formulas for domains with fractal boundary",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/afq7g2q1k1636",
  "venue": "arXiv",
  "versions": []
}