{
  "abstract": "Klurman, Shparlinski and Teräväinen showed that there is a set of at most $(\\log Q)^{1+\\varepsilon}$ primes of polylogarithmic size whose subset products cover the unit group $(\\mathbb{Z}/q\\mathbb{Z})^\\times$ for almost all moduli $q\\le Q$. We show that $(1/\\log2+o(1))\\log Q$ primes suffice, which is optimal to first order, since $m$ primes have at most $2^m$ subset products. The proof combines a doubling argument of Erdős--Rényi type for random subset products in a finite abelian group, modified to tolerate a small set of exceptional characters, with a one-time ``repair'' of the few primitive characters whose $L$-functions have zeros near $s=1$. An exact weighted repair gives a variant with at most $(2/\\log2+o(1))\\log Q$ primes and an exceptional set described explicitly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04678",
  "author_total": 1,
  "authors": [
    {
      "name": "Scott D. Hughes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1ekqdkckv37xm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04678",
  "fulltext_url": null,
  "id": "1ekqdkckv37xm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04678",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.04678",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ekqdkckv37xm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04678",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04678"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Few small primes suffice to cover unit groups",
  "title_text": "Few small primes suffice to cover unit groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/1ekqdkckv37xm",
  "venue": "arXiv",
  "versions": []
}