{
  "abstract": "We test hyperuniformity from one large realization of a stationary point process. Hyperuniformity is equivalent to the average $A_r$ of the structure factor over $B_r$ vanishing as $r\\downarrow0$, so the target is a low-frequency average rather than the value of the structure factor at the origin, which need not exist. We estimate $A_r$ from squared Fourier coefficients at frequencies drawn uniformly from $B_r$; if these coefficients are asymptotically Gaussian at almost every fixed frequency, the squared coefficients converge to independent variables with mean $A_r$. Assuming $A_r=s+c r^α+\\varepsilon_r$ with $|\\varepsilon_r|\\leq Lr^β$ and given exponents $0<α<β$, a two-radius extrapolation removes the $r^α$ term and estimates $s$ with deterministic error of order $r^β$. This gives confidence bounds and a one-sided test of $H_0:s=0$ with asymptotic level at most $γ$, consistent against every fixed $s>0$ as $R\\to\\infty$, then the number of sampled frequencies tends to infinity, and then $r\\downarrow0$. Essentially free translation actions with completely positive entropy satisfy the Fourier assumption.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27961",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Björklund"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ymabveyp62pp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27961",
  "fulltext_url": null,
  "id": "9ymabveyp62pp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27961",
  "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.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214176368",
  "pdf_url": "https://arxiv.org/pdf/2609.27961",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ymabveyp62pp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.27961",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27961"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Randomized Spectral Inference for Hyperuniformity",
  "title_text": "Randomized Spectral Inference for Hyperuniformity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:14Z",
  "url": "https://research.draconitic.com/paper/9ymabveyp62pp",
  "venue": "arXiv",
  "versions": []
}