{
  "abstract": "For an integer $k \\ge 0$, let $f_k(x) = 1 + e(x) + e((k+2)x)$ and $g_k(x) = 1 + e(x) - e((k+2)x)$ on $\\mathbb{T} = \\mathbb{R}/\\mathbb{Z}$, where $e(x) = e^{2πi x}$. Mockenhaupt conjectured that $\\|g_k\\|_{L^p(\\mathbb{T})} > \\|f_k\\|_{L^p(\\mathbb{T})}$ whenever $2k < p < 2k+2$. The conjecture was previously known for $k \\le 5$. We give a single analytic proof valid for every $k \\ge 4$; in particular, this settles all previously open cases $k \\ge 6$ and establishes the conjecture for every $k \\ge 0$. The proof reduces the norm comparison to resonant Fourier coefficients on the two-torus and represents these coefficients, after analytic continuation, by triple-Bessel integrals. Neumann's product formula and the Weber-Schafheitlin formula yield a quantitative positive lower bound for the leading mode, while the remaining odd modes are controlled by a uniform tail estimate. The leading mode is then shown to dominate the tail for every $k \\ge 4$. AI Usage. The mathematical argument of this paper was produced by the auto-research system Apex Math, an AI system built by Apex Intelligence. See Appendix A for the complete AI usage statement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09740",
  "author_total": 6,
  "authors": [
    {
      "name": "Guancheng Pan"
    },
    {
      "name": "Chengsong You"
    },
    {
      "name": "Hengyu Wang"
    },
    {
      "name": "Junwei Zhou"
    },
    {
      "name": "Wenjun Zhang"
    },
    {
      "name": "Yongchao Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/acsnn9hqz4vjs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09740",
  "fulltext_url": null,
  "id": "acsnn9hqz4vjs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09740",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7212164073",
  "pdf_url": "https://arxiv.org/pdf/2609.09740",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/acsnn9hqz4vjs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.09740",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09740"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture",
  "title_text": "Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/acsnn9hqz4vjs",
  "venue": "arXiv",
  "versions": []
}