{
  "abstract": "We introduce lower and upper macroscopic intermediate dimensions of unbounded subsets of Euclidean space by combining covering contributions from all dyadic annuli beyond a given scale. For every unbounded Borel set $E\\subset\\R^d$ and $m\\in \\{1,\\ldots,d-1\\}$, we prove that the macroscopic intermediate dimensions of $P_VE$ equal the corresponding intrinsic capacity profiles for almost every $m$-dimensional linear subspaces $V\\subset \\R^d$. One full-measure set works for both dimensions and every parameter $θ\\in[0,1]$. The proof combines a covering--capacity comparison with an exponent gap, fractional-moment estimates for projected coverings, and capacity-weighted retention of source shells. The dimensions recover macroscopic Hausdorff dimension at parameter zero, are invariant under quasi-isometries, and are continuous at positive parameters. The upper macroscopic intermediate dimension agrees with the upper discrete intermediate dimension introduced in \\cite{LXZ}, whereas the lower one can differ. We compute the projected dimensions of bounded-base discrete digit Moran sets, including the self-similar case obtained when the bases and digit sets are constant, and give applications to Lipschitz graphs and arithmetic sumsets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04395",
  "author_total": 2,
  "authors": [
    {
      "name": "Jinjun Li"
    },
    {
      "name": "Lifeng Xi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29fmts0543fyb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04395",
  "fulltext_url": null,
  "id": "29fmts0543fyb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04395",
  "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"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7220693539",
  "pdf_url": "https://arxiv.org/pdf/2610.04395",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29fmts0543fyb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04395",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04395"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Projection theorems for macroscopic intermediate dimensions",
  "title_text": "Projection theorems for macroscopic intermediate dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/29fmts0543fyb",
  "venue": "arXiv",
  "versions": []
}