{
  "abstract": "We prove that, for every irrational $ξ$, any $F_σ$ additive subgroup of $\\R$ has an $F_σ$ $\\Q$-vector space extension of the same Hausdorff dimension satisfying $A+ξA=\\R$. Iteration gives simultaneous surjectivity for any prescribed countable family of irrational multipliers. Such a space exists in every prescribed dimension $d\\in[0,1)$. We also construct in ZFC a zero-dimensional lightface $Δ^0_3$ $\\Q$-vector subspace $A$ for which $A+ξA=\\R$ holds for every irrational $ξ$, answering both parts of Question~2 of Ye, Yu, and Zhao. This universal group is a difference of two $F_σ$ sets, but is neither $F_σ$ nor $G_δ$. Both constructions use direct inductions on finite binary strings and admit effective versions. As applications, we obtain zero-dimensional $F_σ$ groups whose Cartesian squares have Hausdorff dimension one, and relate their dilate sums to Marstrand's projection theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38970",
  "author_total": 1,
  "authors": [
    {
      "name": "Liang Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9q53mngpd8k2a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38970",
  "fulltext_url": null,
  "id": "9q53mngpd8k2a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38970",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7215046411",
  "pdf_url": "https://arxiv.org/pdf/2609.38970",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9q53mngpd8k2a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38970",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38970"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Small additive groups with $A+ξA=\\mathbb R$",
  "title_text": "Small additive groups with A+ξA=ℝ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/9q53mngpd8k2a",
  "venue": "arXiv",
  "versions": []
}