{
  "abstract": "We introduce a spectrum of dimensions of measures, the \\emph{packing intermediate dimensions} $\\dim_{P,θ}μ$, $θ\\in(0,1]$, of a Borel probability measure $μ$ on $\\mathbb{R}^n$ and the \\emph{quasi-Hausdorff dimension} $\\dim_{qH}μ$. We show that this spectrum encodes the information on $μ$ that governs the packing dimension of its projections and Hölder images. First, $\\dim_{qH}μ$ is the exact threshold for the packing dimension profiles of Falconer and Howroyd: for every real $0<s\\leq n$, $\\dim_P^sμ=s$ if and only if $s\\leq\\dim_{qH}μ$. Second, we prove a lower bound for the profiles in terms of $\\dim_Hμ$, $\\dim_{P,θ}μ$ and the Assouad dimension of $\\operatorname{supp}μ$, which improves the bound of Falconer and Mattila, the best previously known general lower bound for the packing dimension of projected measures. The improvement is strict whenever $\\dim_Hμ<\\dim_Pμ$ and $\\dim_A(\\operatorname{supp}μ)<n$, and when $\\dim_A(\\operatorname{supp}μ)=n$ it occurs exactly when $\\dim_{P,θ}μ$ exceeds an explicit universal lower bound for some $θ$ in an explicit range. We present examples of measures with $\\dim_Hμ<\\dim_Pμ<\\dim_A(\\operatorname{supp}μ)=n$ for which our bound gives the exact value $\\dim_Pμ_V=\\min\\{\\dim_Pμ,m\\}$ for almost every $m$-dimensional subspace $V$, whereas the bound of Falconer and Mattila is strictly smaller. As applications, for a large family of random $α$-Hölder maps $f_ω:\\mathbb{R}^n\\to\\mathbb{R}^m$, we prove that, for compactly supported $μ$, $\\dim_Pμ_{f_ω}=m$ almost surely if and only if $αm\\leq\\dim_{qH}μ$, and that the packing analogue of Kahane's formula holds for fractional Brownian motions if and only if $\\dim_{qH}μ=\\dim_Pμ$. Analogous results are obtained for analytic sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18929",
  "author_total": 1,
  "authors": [
    {
      "name": "Nicolas Angelini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/axmrwysx53112.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18929",
  "fulltext_url": null,
  "id": "axmrwysx53112",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18929",
  "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.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7222006083",
  "pdf_url": "https://arxiv.org/pdf/2610.18929",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/axmrwysx53112.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.18929",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18929"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Packing intermediate dimensions of measures and applications",
  "title_text": "Packing intermediate dimensions of measures and applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/axmrwysx53112",
  "venue": "arXiv",
  "versions": []
}