{
  "abstract": "Let $q=2^n$ with $n\\ge6$, and let $\\mathbb{F}_q$ be the finite field with $q$ elements. For a subspace $E$ of $\\mathbb{F}_q$, define \\[ S(E)=\\sum_{x\\in E\\setminus\\{0\\}}x^{-1}.\\] Let $N_{n,k}$ denote the number of $k$-dimensional $\\mathbb{F}_2$-subspaces $E$ for which $S(E)=0$. We prove that, if $k\\ge3$ and $n\\ge2k+1$, then \\[ \\left|\\frac{2^nN_{n,k}}{\\genfrac{[}{]}{0pt}{}{n}{k}_{2}}-1\\right|<2^{r^2+r}2^{n(1-r/2)},\\] where $r=\\lfloor\\frac{k-1}{2}\\rfloor$, and $\\genfrac{[}{]}{0pt}{}{n}{k}_{2}$ is the Gaussian binomial coefficient. In particular, \\[ N_{n,k}\\sim2^{-n}\\genfrac{[}{]}{0pt}{}{n}{k}_{2}\\quad (n\\to\\infty)\\] uniformly for $n/3<k\\le(n-1)/2$. Combined with the known low-dimensional cases and the symmetry between dimensions $k$ and $n-k$, and the elementary middle-dimensional subfield construction, this proves a conjecture of Carlet: for every $3\\le k\\le n-3$, there exists a $k$-dimensional $\\mathbb{F}_2$-subspace $E$ such that $S(E)=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06445",
  "author_total": 1,
  "authors": [
    {
      "name": "Kaimin Cheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0jbxjr0w5b6vs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06445",
  "fulltext_url": null,
  "id": "0jbxjr0w5b6vs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06445",
  "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",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7220652083",
  "pdf_url": "https://arxiv.org/pdf/2610.06445",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0jbxjr0w5b6vs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06445",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06445"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counting zero-sum subspaces for the multiplicative inverse function",
  "title_text": "Counting zero-sum subspaces for the multiplicative inverse function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/0jbxjr0w5b6vs",
  "venue": "arXiv",
  "versions": []
}