{
  "abstract": "For every finite $P\\subset\\mathbb{R}^2$ we prove $|\\{p\\cdot q: p,q\\in P\\}|\\gg |P|^{199/295}$, with an absolute constant and no logarithmic loss, where $199/295 = 2/3+7/885$. This improves the bound $2/3+7/1425$ of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Newton, and Senger. The main new ingredient is the inequality $|F^{(2)}G^{(2)}/(F^{(2)}G)| \\le |AB|(|AF|/|A|)^4(|BG|/|B|)^3$: each radial profile retains its own Petridis growth-minimizing subset, and the product of the two subsets serves as a common base for both growth operators; a prime-power construction shows this is sharp under its hypotheses. The second ingredient is a weighted-median replacement for the dyadic selection in the squeezing argument of Roche-Newton and Wong, upgrading their seven-factor expander to a logarithm-free form; an appendix gives the complete proof from the Solymosi-Zahl incidence theorem. We complement the lower bound with a third-moment structure theorem for near-extremal configurations and a construction showing that the pinned problem admits no linear lower bound: $\\max_{p\\in P}|p\\cdot P| = O(|P|/\\sqrt{\\log |P|})$ is attainable. The algebraic core, the median lemma, the surface identities, and the pinned construction are formally verified in Lean 4 and included as ancillary files.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06730",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhipeng Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/85ajamneenywp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06730",
  "fulltext_url": null,
  "id": "85ajamneenywp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06730",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212055272",
  "pdf_url": "https://arxiv.org/pdf/2609.06730",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/85ajamneenywp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06730",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06730"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A dot-product bound from separate growth-minimizing bases",
  "title_text": "A dot-product bound from separate growth-minimizing bases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/85ajamneenywp",
  "venue": "arXiv",
  "versions": []
}