{
  "abstract": "Let $G$ be a countable locally finite abelian group, and let \\[ G_1\\leq G_2\\leq\\cdots, \\qquad \\bigcup_{i\\geq1}G_i=G, \\] be any filtration of $G$ by finite subgroups. For $A\\subseteq G$, let $\\mathcal P(A)$ denote the set of all finite subset sums of elements of $A$, and let $2G:=\\{2g:g\\in G\\}$. We prove that $|2G|=\\infty$ if and only if for every filtration and every interval $[α,β]\\subseteq[0,1]$, there exists $A\\subseteq G$ such that the set of limit points of \\[ \\left( \\frac{|\\mathcal P(A)\\cap G_i|}{|G_i|} \\right)_{i\\geq1} \\] is exactly $[α,β]$. We also prove a positive-density result that does not require $|2G|=\\infty$: if $G$ is infinite and $\\mathcal P(A)$ has positive upper density along the given filtration, then there is an infinite set $B\\subseteq \\mathcal P(A)$ such that $B+B\\subseteq \\mathcal P(A)$. Combining this with the realization theorem, we show that, when $|2G|=\\infty$, every interval $[α,β]$ with $0\\leqα\\leqβ\\leq1$ and $β>0$ can be realized by a set $A$ with this additional property.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14630",
  "author_total": 3,
  "authors": [
    {
      "name": "Norbert Hegyvári"
    },
    {
      "name": "Thang Pham"
    },
    {
      "name": "Boqing Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0863ytbnjy2dm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14630",
  "fulltext_url": null,
  "id": "0863ytbnjy2dm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14630",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7213262532",
  "pdf_url": "https://arxiv.org/pdf/2609.14630",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0863ytbnjy2dm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14630",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14630"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Subset-Sum Density Realization in Locally Finite Abelian Groups",
  "title_text": "Subset-Sum Density Realization in Locally Finite Abelian Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/0863ytbnjy2dm",
  "venue": "arXiv",
  "versions": []
}