{
  "abstract": "The classical Thompson's conjecture asserts that in any finite nonabelian simple group $G$, there exists a conjugacy class $S$ whose square is equal to the entire group. Shalev (Annals Math., 2009) conjectured that a much stronger quantitative version holds: there exists $ε> 0$ such that for any conjugacy class $S$ with $|S| \\geq |G|^{1-ε}$, we have $G \\setminus \\{1\\} \\subseteq S^2$. While the conjecture turns out to fail in general, we prove a close variant of the conjecture over $\\operatorname{PSL}_n(2)$: There exist $ε>0$ and $N \\in \\mathbb{N}$ such that for all $n\\geq N$, every conjugacy class $S\\subseteq\\operatorname{PSL}_n(2)$ with $|S|\\geq|\\operatorname{PSL}_n(2)|^{1-ε}$ satisfies $\\{x\\in\\operatorname{PSL}_n(2):\\operatorname{supp}(x)\\neq1\\}\\subseteq SS^{-1}.$ On the other hand, we show that for each $ε>0$, there exist conjugacy classes $S \\subset \\operatorname{PSL}_n(8)$ of size $>|\\operatorname{PSL}_n(8))|^{1-ε}$ for which $SS^{-1}$ misses even elements with support size linear in $n$. In addition, for every odd $n \\ge 5$, we construct an inverse-closed normal set in $\\operatorname{PSL}_n(5)$ with fixed positive density whose square misses a nonidentity element. This answers on the negative a question of Shalev (ECM, 2021) and disproves a conjecture of Pyber (Kourovka notebook, 2022).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22004",
  "author_total": 1,
  "authors": [
    {
      "name": "Ohad Sheinfeld"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bx9sbj8x95507.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22004",
  "fulltext_url": null,
  "id": "bx9sbj8x95507",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22004",
  "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.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7213893043",
  "pdf_url": "https://arxiv.org/pdf/2609.22004",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bx9sbj8x95507.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.22004",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22004"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Shalev's Quantitative Version of Thompson's Conjecture",
  "title_text": "On Shalev's Quantitative Version of Thompson's Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/bx9sbj8x95507",
  "venue": "arXiv",
  "versions": []
}