{
  "abstract": "We develop a geometric lifting method for inverse sumset problems in compact connected abelian groups. The first result is an analogue of Freiman's $3k-4$ theorem, that is every compact set $A\\subseteq G$ of sufficiently small Haar measure satisfying $μ_G(A+A)<3μ_G(A)$ is contained in a one dimensional Bohr set of measure at most $μ_G(A+A)-μ_G(A)$. This resolves a question of Christ and Iliopoulou. The proof combines Bilu's theorem with a geometric refinement of the spillover argument. We also establish a sharp projection theorem. Under a continuous surjective homomorphism with connected kernel, a compact set of sufficiently small positive measure and doubling at most $K$, where $2\\le K<3$, has image of doubling at most $2K-2$, and this factor is best possible. Further consequences include variants of the $3k-4$ theorem for popular sumsets and an inverse theorem for Tao's convolution inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27322",
  "author_total": 2,
  "authors": [
    {
      "name": "Yifan Jing"
    },
    {
      "name": "Yuchen Meng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3t6sm9xrj0hy8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27322",
  "fulltext_url": null,
  "id": "3t6sm9xrj0hy8",
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  "landing_url": "https://arxiv.org/abs/2609.27322",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "arxiv",
      "math.CA"
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      "arxiv",
      "math.CO"
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      "arxiv",
      "math.GR"
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  "openalex": "W7214224914",
  "pdf_url": "https://arxiv.org/pdf/2609.27322",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3t6sm9xrj0hy8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27322",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27322"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Geometric lifting and Freiman's $3k-4$ theorem in compact connected abelian groups",
  "title_text": "Geometric lifting and Freiman's 3k−4 theorem in compact connected abelian groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/3t6sm9xrj0hy8",
  "venue": "arXiv",
  "versions": []
}