{
  "abstract": "We investigate the spectral properties of a class of inhomogeneous self-similar measures, which does not admit a non-trivial infinite convolution structure. A central example is the golden-mean self-similar measure $μ$, for which the existence of an exponential orthonormal basis in the associated $L^2$-space has remained a long-standing open problem. The usual approach for homogeneous self-similar measures does not apply here, new methods are required. We establish several basic properties of the measure and then carry out a detailed numerical study of the zero set of its Fourier transform. Using a scanning and refinement algorithm that combines uniform grid sampling, quadratic interpolation, and golden-section search, we examine a wide range and find no real zeros of $\\widehatμ$, which provides concrete evidence that $μ$ is very likely non-spectral, suggesting that inhomogeneity may serve as a natural obstruction to the existence of exponential orthonormal bases. To the best of our knowledge, our paper is the first attempt to study the spectrality of such measures through a combined analytic and numerical framework.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04701",
  "author_total": 2,
  "authors": [
    {
      "name": "Yi-Qiu Mao"
    },
    {
      "name": "Zhi-Yi Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7xag2g9qp79kp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04701",
  "fulltext_url": null,
  "id": "7xag2g9qp79kp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04701",
  "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"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7211915314",
  "pdf_url": "https://arxiv.org/pdf/2609.04701",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7xag2g9qp79kp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04701",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04701"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the spectrality of the non-homogeneous golden-mean self-similar measure",
  "title_text": "On the spectrality of the non-homogeneous golden-mean self-similar measure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/7xag2g9qp79kp",
  "venue": "arXiv",
  "versions": []
}