{
  "abstract": "We consider the problem of finding lower bounds for integer linear combinations of $\\sqrt{a_1},\\ldots,\\sqrt{a_K}$, where $a_1,\\ldots,a_K$ are positive integers such that their square roots are linearly independent over the rationals. We use a probabilistic approach and prove that for $K\\geq 8$ and nonzero integers $m_1,\\ldots,m_K$, $$ \\bigg|\\sum_{n\\leq K} m_n\\sqrt{a_n}\\bigg| > e^{\\frac{2^{K-1}-1}{K} - \\frac{1}{2}} \\bigg(\\max_{n\\leq K}|m_n|\\sqrt{a_n}\\cdot \\sqrt{K}\\bigg)^{-(2^{K-1}-1)}. $$ This inequality improves the dependence on $K$ in the classical product bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14161",
  "author_total": 4,
  "authors": [
    {
      "name": "Marco Aymone"
    },
    {
      "name": "Samuel Figueredo"
    },
    {
      "name": "Siddharth Iyer"
    },
    {
      "name": "Christian Táfula"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/639yx94xapddj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14161",
  "fulltext_url": null,
  "id": "639yx94xapddj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14161",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213339423",
  "pdf_url": "https://arxiv.org/pdf/2609.14161",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/639yx94xapddj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.14161",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14161"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Quantitative linear independence for square roots",
  "title_text": "Quantitative linear independence for square roots",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/639yx94xapddj",
  "venue": "arXiv",
  "versions": []
}