{
  "abstract": "We obtain a uniform pointwise estimate for short Weyl sums of degree $n\\geq3$ in the intermediate rational-approximation range $$ \\frac{1}{qx^{n-2}y}\\ll|λ| \\ll\\frac{1}{qy^{n-1}}. $$ This range arises from a second application of Dirichlet's rational approximation theorem in estimating the residual integral that occurs in the derivation of an asymptotic formula for Waring's problem with almost proportional summands. For $r=2^n+1$ and fixed positive numbers $μ_1,\\ldots,μ_r$ satisfying $$ μ_1+\\cdots+μ_r=1, $$ we derive an asymptotic formula for the number of representations $$ x_1^n+\\cdots+x_r^n=N, \\qquad |x_i^n-μ_iN|\\leq H,\\qquad 1\\le i \\le r, $$ valid for $$ N^{1-θ(n,r)+\\varepsilon}\\le H \\le \\frac{N}{\\ln N}, \\quad θ(n,r)= \\frac{2}{n\\bigl((r-1)(n-1)+2\\bigr)}. $$ The resulting admissible lower bound for $H$ improves the previously known bound for every $n\\geq3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25787",
  "author_total": 1,
  "authors": [
    {
      "name": "Karimjon Ibrohimjonovich Mirzoabdughafurov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2hc1gfba0f9f0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25787",
  "fulltext_url": null,
  "id": "2hc1gfba0f9f0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25787",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204488610",
  "pdf_url": "https://arxiv.org/pdf/2608.25787",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2hc1gfba0f9f0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.25787",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25787"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Intermediate-Range Estimates for Short Weyl Sums and Waring's Problem with Almost Proportional Summands",
  "title_text": "Intermediate-Range Estimates for Short Weyl Sums and Waring's Problem with Almost Proportional Summands",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/2hc1gfba0f9f0",
  "venue": "arXiv",
  "versions": []
}