{
  "abstract": "Let $A\\subset\\mathbb{R}^d$, $d\\geq3$, be the attractor of a self-similar IFS whose rotations generate a dense subgroup of $\\mathrm{SO}(d)$. For every $1\\leq k<\\dim_{\\mathrm{H}} A$ and $V\\in\\mathrm{Gr}(d,k)$, we prove that $π_VA$ has positive $k$-dimensional Lebesgue measure uniformly in $V$, and $\\dim_{\\mathrm{H}}(A\\capπ_V^{-1}\\{y\\})=\\dim_{\\mathrm{H}} A-k$ for Lebesgue-almost every $y\\inπ_VA$. Moreover, if $μ$ is a self-similar measure for such an IFS with $I_t(μ)<\\infty$ for some $k<t<d$, then every $μ_V=(π_V)_*μ$ is equivalent to the $k$-dimensional Lebesgue measure restricted to $π_VA$, with density $f_V$ satisfying $\\sup_V\\int_{\\{f_V>M\\}}f_V\\,d\\mathcal{L}_V^k\\lesssim e^{-c(\\log M)^{1/3}}$. Under strong separation, the conditional measures on the fibres are exact dimensional of dimension $\\dim_{\\mathrm{H}}μ-k$. The proof uses self-similarity to iterate $L^1$ norms of differences of the Gaussian-smoothing of $μ_V$ between scales. Unlike $L^q$, $q>1$, the $L^1$ norm does not grow under rescaling, allowing Varjú's $L^2$ weak gap theorem under dense support condition to be used, and avoiding to use the much stronger uniform spectral gap assumption.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29301",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiong Jin"
    },
    {
      "name": "Tuomas Sahlsten"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0wwwnf6ayejjt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29301",
  "fulltext_url": null,
  "id": "0wwwnf6ayejjt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29301",
  "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.CA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214328249",
  "pdf_url": "https://arxiv.org/pdf/2609.29301",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0wwwnf6ayejjt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.29301",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29301"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Absolute continuity of projections and dimension conservation for self-similar sets and measures",
  "title_text": "Absolute continuity of projections and dimension conservation for self-similar sets and measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/0wwwnf6ayejjt",
  "venue": "arXiv",
  "versions": []
}