{
  "abstract": "We establish sharp mean value estimates for various exponential sums lacking translation-dilation invariance properties. For example, suppose that $k\\ge 1$ and $s>0$. Then, for each $\\varepsilon>0$, one has \\[ \\int_{[0,1)^k}\\biggl| \\sum_{1\\le h\\le X}\\sum_{1\\le z\\le X}e(hzα_1+\\ldots +hz^kα_k)\\biggr|^{2s}\\,{\\rm d}\\boldsymbol α\\ll X^\\varepsilon\\bigl( X^s+X^{2s-k(k+3)/2}\\bigr), \\] confirming the main conjecture for a new family of exponential sums. Our methods are based on the application of very recent work of the author employing the nested efficient congruencing method.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05525",
  "author_total": 1,
  "authors": [
    {
      "name": "Trevor D. Wooley"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1kb48zqed0gq4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05525",
  "fulltext_url": null,
  "id": "1kb48zqed0gq4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05525",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220663827",
  "pdf_url": "https://arxiv.org/pdf/2610.05525",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1kb48zqed0gq4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05525",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05525"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sharp mean value estimates in the absence of translation-dilation invariance",
  "title_text": "Sharp mean value estimates in the absence of translation-dilation invariance",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/1kb48zqed0gq4",
  "venue": "arXiv",
  "versions": []
}