{
  "abstract": "Let $D\\subset\\Bbb Z$ with cardinality $q\\ge 2$, and let $b\\in \\Bbb Z$ with $q<b$, and let $μ:=μ_{b,D}$ be the associated self-similar measure. It is well known that if there exists $L\\subset \\Bbb Z$ such that $(b,D,L)$ be a Hadamard triple, then the Beurling dimension of the spectrum of $μ$ exhibits an intermediate structural property. In this paper, we establish a stronger result that both Beurling dimension and Beurling density of the spectra of $μ$ can achieve full flexibility simultaneously. More precisely, for any $t\\in(0, \\frac{\\log q}{\\log b})$ and $s\\in [0,\\infty]$, there exists a spectrum $Λ:=Λ_{t,s}$ of $μ$ such that $$\\dim_{Be}(Λ)=t,\\quad D_t^+ (Λ)=s.$$ Here, $\\dim_{Be}$ and $D_t^+$ denote the Beurling dimension and the $t$-Beurling density, respectively. We further prove that the set of such spectrum whose Beurling dimension and Beurling density are equal to any fixed $t$ and $s$ has the cardinality of the continuum. \\par This work generalizes a previous result of Lu \\cite{Lu}, answers an open question raised by Dai, Fu and He \\cite[Conjecture 5.3]{DaiFuHe}, and sheds new light on the fine structural properties of spectra for singularly continuous spectral measures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29660",
  "author_total": 2,
  "authors": [
    {
      "name": "Zong-Sheng Liu"
    },
    {
      "name": "Xiao-Yu Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dk58x1k76e1bt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29660",
  "fulltext_url": null,
  "id": "dk58x1k76e1bt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29660",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7204929656",
  "pdf_url": "https://arxiv.org/pdf/2608.29660",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dk58x1k76e1bt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.29660",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29660"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Beurling density of the spectrum of self-similar measure generated by Hadamard triple",
  "title_text": "The Beurling density of the spectrum of self-similar measure generated by Hadamard triple",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/dk58x1k76e1bt",
  "venue": "arXiv",
  "versions": []
}