{
  "abstract": "We provide several results showing that sums of ``small'' Cantor sets can turn into an interval or a finite union of intervals. Specifically, we show that 1) There exists a Cantor set $C\\subset [0,1]$ of zero logarithmic capacity such that $C+C=[0,2]$. 2) For any $n\\ge 2$ there exists a Cantor set $C\\subset [0,1]$ of zero logarithmic capacity such that a Minkowski sum of $n$ copies of $C$ is an interval $[0,n]$, while a sum of any smaller number of copies of $C$ is still a Cantor set. 3) For any $n\\ge 2$ there exists a collection of Cantor sets $C_1, C_2, \\ldots, C_n\\subset [0,1]$ of zero lower box-counting dimension such that a Minkowski sum of any $n-1$ sets from this collection is a Cantor set of zero lower box-counting dimension, while the sum of all $n$ sets is an interval $[0,n]$. 4) There is an uncountable family of Cantor sets of zero lower box-counting dimension such that the sum of any two different sets from this family is a finite union of intervals. 5) For each of the properties above, there are examples of Cantor sets with the same property having zero Hausdorff $h$-measure for an arbitrary given gauge function $h$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17005",
  "author_total": 3,
  "authors": [
    {
      "name": "Anton Gorodetski"
    },
    {
      "name": "Alexandro Luna"
    },
    {
      "name": "William Wood"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/85m5p7xec9czw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17005",
  "fulltext_url": null,
  "id": "85m5p7xec9czw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17005",
  "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.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221783220",
  "pdf_url": "https://arxiv.org/pdf/2610.17005",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/85m5p7xec9czw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17005",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17005"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "On Sums of \"Small\" Cantor Sets",
  "title_text": "On Sums of \"Small\" Cantor Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/85m5p7xec9czw",
  "venue": "arXiv",
  "versions": []
}