{
  "abstract": "Let $μ$ be an $n$-AD-regular measure in $\\mathbb{R}^d$. Chousionis, Garnett, Le and Tolsa [CGLT] proved that $μ$ is uniformly $n$-rectifiable if and only if the square function built from the density differences $Δ_μ(x,r)=μ(B(x,r))/r^n-μ(B(x,2r))/(2r)^n$ satisfies a Carleson condition. In this paper we show that the same characterization holds if the density is first composed with a function $F$ which is bi-Lipschitz on the interval $[c_0^{-1},c_0]$ determined by the AD-regularity constant $c_0$. The main example is $F=\\log$, introduced in [Le], for which the square function takes the scale-invariant form $Δ_μ^{\\log}(x,r) = \\log\\bigl(μ(B(x,r))/μ(B(x,2r))\\bigr)+n\\log 2$. We give a complete proof, extend the statement to the smooth square functions of [CGLT], where the density is replaced by the convolution of $μ$ with a Gaussian or a more general radial kernel, discuss what happens when $F$ is not bi-Lipschitz, and treat the case $μ(\\mathbb{R}^d)<\\infty$, where the behavior of $F$ near zero enters in only one of the two implications. We also show that the qualitative characterization of $n$-rectifiable measures by Tolsa and Toro [TT], in terms of the same square function at $μ$-almost every point, holds after composition with any locally bi-Lipschitz $F$. This requires neither AD-regularity nor doubling, and for $F=\\log$ the condition $\\lim_{r\\to0}Δ_μ(x,r)=0$ becomes $\\lim_{r\\to0}μ(B(x,r))/μ(B(x,2r))=2^{-n}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30166",
  "author_total": 1,
  "authors": [
    {
      "name": "Triet M. Le"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24az6c6z1pa41.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30166",
  "fulltext_url": null,
  "id": "24az6c6z1pa41",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30166",
  "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"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7204944244",
  "pdf_url": "https://arxiv.org/pdf/2608.30166",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24az6c6z1pa41.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30166",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30166"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Square Functions and Rectifiability under Monotone Transformations of the Density",
  "title_text": "Square Functions and Rectifiability under Monotone Transformations of the Density",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/24az6c6z1pa41",
  "venue": "arXiv",
  "versions": []
}