{
  "abstract": "Given $0\\le m<n$ and a positive Radon measure $μ$ on $\\mathbb R^n$ with at most polynomial growth, assume that it has finite $m$-dimensional Gaussian discrepancy, defined as $$\\mathcal D(μ):=\\limsup_{R\\to\\infty}\\Bigl[R^2\\sup_{x\\in\\mathrm{spt}μ}|G_R^μ(x)-1|\\Bigr],$$ where $G_R^μ(x):=(4πR^2)^{-m/2}\\int_{\\mathbb R^n} e^{-|z-x|^2/(4R^2)}\\,dμ(z)$. Provided that $μ$ admits a flat blow-down $\\mathcal H^m\\llcorner V$ (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) $0\\leδ<\\infty$, the so-called width of $μ$, such that $$\\mathrm{spt}μ\\subseteq \\bar B_δ(V+x_0),\\quad δ^2\\le C(m,n)\\mathcal{D}(μ),$$ where $\\bar B_δ(V+x_0)$ denotes the closed $δ$-neighborhood of a suitable translate of $V$. Note that the exponent is optimal because of scale invariance. In particular, when $\\mathcal D(μ)=0$, the support of $μ$ is included in an $m$-dimensional affine plane. We use this fact to give a simpler proof of Preiss's rectifiability theorem: while still based on Gaussian averages of $μ$, this new proof avoids showing uniqueness (and hence conicality) of blow-downs of $m$-uniform measures, as well as studying Gaussian-weighted moments beyond second order. The strategy of proof consists of an anisotropic blow-down of $μ$, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20385",
  "author_total": 1,
  "authors": [
    {
      "name": "Alessandro Pigati"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/910fvx4xyt8th.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20385",
  "fulltext_url": null,
  "id": "910fvx4xyt8th",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20385",
  "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.CA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7221954710",
  "pdf_url": "https://arxiv.org/pdf/2610.20385",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/910fvx4xyt8th.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.20385",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20385"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Gaussian rigidity and width of measures",
  "title_text": "Gaussian rigidity and width of measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/910fvx4xyt8th",
  "venue": "arXiv",
  "versions": []
}