{
  "abstract": "$\\newcommand{\\X}{\\mathbb{X}}\\newcommand{\\SC}{\\mathcal{C}}$ We establish the existence of a \"maximally spread out\" Borel probability measure on a totally bounded subset $\\SC$ of a (quasi)-Banach space $\\X$ under two mild conditions: (i) a growth condition on the covering numbers $N(\\SC, ε)$ of $\\SC$, and (ii) a technical topological condition that is in particular satisfied whenever $\\SC\\subset\\X$ is closed, bounded, and convex. More formally, condition (i) requires that the so-called lower power-exponential Minkowski dimension of $\\SC$, i.e., \\[s_\\ast:=\\liminf_{ε\\downarrow 0}\\frac{\\log\\log N(\\SC,ε)}{\\log(1/ε)}\\] satisfies $s_\\ast>0$. Under these conditions, we construct a Borel probability measure $μ$ on $\\X$ that is critical for $\\SC$, or maximally spread out, meaning that the associated outer measure $μ^\\ast$ satisfies $μ^\\ast(\\X\\setminus\\SC)= 0$ and furthermore satisfies for every $0<s<s_\\ast$ the small-ball condition \\[μ^\\ast(B(x,r))\\le\\exp\\bigl(-c(s)\\cdot(1/r)^s\\bigr)\\quad\\text{ for all }x\\in\\X\\text{ and }0<r<r_0(s).\\] The existence of such a critical measure in particular implies that the so-called power-exponential Hausdorff dimension of $\\SC$ introduced in [J.~Topol.~Anal.~4(2):203--235, 2012] coincides with the lower power-exponential Minkowski dimension. Previous work [Found.~Comput.~Math.~23(1):329--392, 2023] shows that such a critical measure gives rise to a phase transition regarding lossy compression and approximation by quantized neural networks of elements of $\\SC$, provided that the $\\liminf$ in the definition of $s_\\ast$ exists as an actual limit. There, critical measures were constructed for unit balls of certain Besov and Sobolev spaces considered as subsets of $L^2$. In contrast, our construction is completely general. In particular, our results apply to function spaces of dominating mixed smoothness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30549",
  "author_total": 3,
  "authors": [
    {
      "name": "Hannes Matt"
    },
    {
      "name": "Erwin Riegler"
    },
    {
      "name": "Felix Voigtlaender"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3egb2e6kpr7j6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30549",
  "fulltext_url": null,
  "id": "3egb2e6kpr7j6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30549",
  "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"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7204870239",
  "pdf_url": "https://arxiv.org/pdf/2608.30549",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3egb2e6kpr7j6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30549",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30549"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Maximally Spread Out Measures and Implications for Phase Transitions in Approximation Theory",
  "title_text": "Maximally Spread Out Measures and Implications for Phase Transitions in Approximation Theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/3egb2e6kpr7j6",
  "venue": "arXiv",
  "versions": []
}