{
  "abstract": "Let $A\\subset\\mathbb{R}$ be a finite set. Combining the multiplicative slope estimate of Rudnev--Stevens, Cushman's higher-energy regularization, Shakan's $d^+$--$d^\\times$ decomposition, and Solymosi's classical sum--product estimate, we prove \\[ |AA|^{204}|A+A|^{301}\\gtrsim |A|^{675}, \\] where $\\gtrsim$ suppresses a fixed polylogarithmic factor in $|A|$. Consequently, for every $\\varepsilon>0$, \\[ \\max\\{|A+A|,|AA|\\}\\gg_\\varepsilon |A|^{135/101-\\varepsilon}. \\] The proof is organized around two intermediate estimates. For every nonempty finite set $B\\subset\\mathbb{R}_{>0}$, \\[ d^\\times(B)|BB|^{12}|B+B|^{16}\\gtrsim |B|^{38}, \\] whereas for every nonempty finite set $U\\subset\\mathbb{R}$, \\[ d^+(U)^{17}|U+U|^{29}\\gtrsim |U|^{46}. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25711",
  "author_total": 3,
  "authors": [
    {
      "name": "Kaiqiang Zhang"
    },
    {
      "name": "Yuyu Wang"
    },
    {
      "name": "Yuanguo Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/anxq0wq4ewpqw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25711",
  "fulltext_url": null,
  "id": "anxq0wq4ewpqw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25711",
  "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": "W7214081173",
  "pdf_url": "https://arxiv.org/pdf/2609.25711",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/anxq0wq4ewpqw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25711",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25711"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Refined Sum-Product Estimate via Higher Energies",
  "title_text": "A Refined Sum-Product Estimate via Higher Energies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/anxq0wq4ewpqw",
  "venue": "arXiv",
  "versions": []
}