{
  "abstract": "Let $F$ be a function field over an algebraically closed field $K$ and $S$ a finite dimensional $K$-subspace of $F$. The square of $S$ is spanned by all products of pairs of elements in $S$. We conjecture that if $\\dim S^2 \\leq 3 \\dim S - 4$, then $S^2$ must contain a Riemann--Roch space of dimension at least $2 \\dim S - 1 + g$, where $g$ is the genus of $F$. This generalizes a theorem of Freiman from additive combinatorics, stating that small sumsets must contain long arithmetic progressions. We prove our conjecture in the case that $S$ is contained in a Riemann--Roch space of dimension at most $3/2\\dim S + 1$. For the proof we study the annihilator of $S$ and introduce the notion of weight for linear forms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06822",
  "author_total": 2,
  "authors": [
    {
      "name": "Nihan Tanısalı"
    },
    {
      "name": "Mieke Wessel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9rvwqmfcvm3h4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06822",
  "fulltext_url": null,
  "id": "9rvwqmfcvm3h4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06822",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212079148",
  "pdf_url": "https://arxiv.org/pdf/2609.06822",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9rvwqmfcvm3h4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06822",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06822"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "High dimensional Riemann--Roch spaces in linear spaces with small squares",
  "title_text": "High dimensional Riemann--Roch spaces in linear spaces with small squares",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/9rvwqmfcvm3h4",
  "venue": "arXiv",
  "versions": []
}