{
  "abstract": "Let a=\\{a_n\\} be a non-increasing sequence of positive real numbers adding up to $1$. Cutting out of the unit interval disjoint open gaps of lengths $a_1,a_2,\\dots$, one for each term, leaves a compact set of Lebesgue measure zero, and since the gaps can be placed in infinitely many ways, there are infinitely many such sets. We call them the \\emph{complementary sets} of $a$ and write $\\mathfrak{C}_a$ for the family of all of them. Every set in $\\mathfrak{C}_a$ is compact and totally disconnected, and all of them share the same gap lengths; what distinguishes one from another is only the arrangement of the gaps. Among them, the perfect sets are exactly the Cantor sets with those gaps, and they form a distinguished subclass. The central question of this survey is what such a rearrangement can change and what it cannot. The answer depends sharply on the notion of size: the box dimensions are constant on $\\mathfrak{C}_a$, while the Hausdorff dimension attains every value in a closed interval whose endpoints are realised by two distinguished members, the Cantor set $C_a$ built by placing the gaps in a Cantor-like pattern, and the countable decreasing rearrangement $D_a$. We survey how this dichotomy persists, and how the roles of $C_a$ and $D_a$ are exchanged, across the intermediate dimensions of Falconer, Fraser and Kempton, the Assouad-type and $Φ$-dimensions, and the Hausdorff and packing $h$-measures associated with a general dimension function; we also describe the almost sure behaviour of a randomly rearranged complementary set, where a sharp threshold in the depth of the scales examined separates two incompatible regimes. A final part restricts attention to the Cantor subclass and turns to the dynamical description of the sets C_a.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28960",
  "author_total": 3,
  "authors": [
    {
      "name": "Nicolás Angelini"
    },
    {
      "name": "Carlos Cabrelli"
    },
    {
      "name": "Ursula Molter"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d4m6bsa19wt22.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28960",
  "fulltext_url": null,
  "id": "d4m6bsa19wt22",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28960",
  "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"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223242285",
  "pdf_url": "https://arxiv.org/pdf/2610.28960",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d4m6bsa19wt22.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.28960",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28960"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Same Gaps, Different Sets: A Survey on Complimentary sets and their Dimensions",
  "title_text": "Same Gaps, Different Sets: A Survey on Complimentary sets and their Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/d4m6bsa19wt22",
  "venue": "arXiv",
  "versions": []
}