{
  "abstract": "A discrete set $Λ$ is called a {\\it spectrum} of a Borel probability measure $μ$ if the exponential functions $\\{e^{2πi \\langleλ, x\\rangle}:λ\\inΛ\\}$ form an orthonormal basis for $L^2(μ)$. In this work, we first give necessary and sufficient conditions for a self-replicating translation set to be a spectrum of the self-similar measure associated with a product-form Hadamard triple, which were discovered by the first-named author and Lai [Adv. Math. 431 (2023) Paper No.109257]. These results extend the studies by Łaba and Wang [J. Funct. Anal. 193 (2002) 409-420], Dutkay and Lai [J. Math. Pures Appl. 107 (2017) 183-204] in $\\mathbb R$. As an application, we can explicitly construct such a self-replicating spectrum. Finally, we prove that the dual spectral set conjecture holds for a self-replicating translation set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05830",
  "author_total": 3,
  "authors": [
    {
      "name": "Li-Xiang An"
    },
    {
      "name": "Yan-Song Fu"
    },
    {
      "name": "Ming-Xuan Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5yrsk6w4xd4fn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05830",
  "fulltext_url": null,
  "id": "5yrsk6w4xd4fn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05830",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7211990679",
  "pdf_url": "https://arxiv.org/pdf/2609.05830",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5yrsk6w4xd4fn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05830",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05830"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Characterization on spectra of self-similar measures and the dual spectral set conjecture",
  "title_text": "Characterization on spectra of self-similar measures and the dual spectral set conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/5yrsk6w4xd4fn",
  "venue": "arXiv",
  "versions": []
}