{
  "abstract": "We examine two questions on exceptional orthogonal projections for Assouad dimension. Let $1\\le k<d$ and $1\\le q<d$. For a set $F$ contained in a $q$-dimensional subspace $W\\subset\\R^d$ and containing a nonempty relatively open subset of $W$, we compute both the Hausdorff and packing dimensions of the exceptional $k$-planes as $k(d-k)-|q-k|-1$. In particular, a square in $\\R^3$ has a one-dimensional set of exceptional plane projections. This disproves the bound $k(d-1-k)$ in Question 17.8.1 of Fraser's book as stated, for every $2\\le k<d$. The planar packing question, Question 17.8.2, is already answered affirmatively by Wu's countability theorem. Using that theorem, we also describe the distinction between planar sets of Assouad dimension at most $1$ and those of dimension greater than $1$ when they are embedded in higher-dimensional spaces. The computations use elementary Grassmannian geometry; we make no claim of a new planar projection theorem or of an optimal bound for arbitrary higher-dimensional sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23295",
  "author_total": 1,
  "authors": [
    {
      "name": "Bilel Selmi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4pbracr7b431e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23295",
  "fulltext_url": null,
  "id": "4pbracr7b431e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23295",
  "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.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7222937895",
  "pdf_url": "https://arxiv.org/pdf/2610.23295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4pbracr7b431e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Orthogonal Projections and Assouad Dimension: Resolving Fraser's Questions",
  "title_text": "Orthogonal Projections and Assouad Dimension: Resolving Fraser's Questions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/4pbracr7b431e",
  "venue": "arXiv",
  "versions": []
}