{
  "abstract": "We answer, in the trace metric, a question of Kowalski and Untrau on Wasserstein distances of all orders between the distribution $μ_q$ of the normalized Kloosterman sums $\\mathrm{Kl}(a;q)$, $a\\in\\mathbb{F}_q^\\times$, and the Sato-Tate law $μ_{\\mathrm{ST}}$: we prove bounds for every order $s$ and show that their exponents are optimal for Katz-type moment bounds. In the trace metric we prove $W_s(μ_q,μ_{\\mathrm{ST}})\\ll_s q^{-1/3}$ for $1\\le s<3$, $W_3\\ll q^{-1/3}(\\log q)^{1/3}$, $W_s\\ll_s q^{-1/4-1/(4s)}$ for $3<s<\\infty$ and $W_\\infty\\ll q^{-1/4}$. In the angle metric we obtain bounds of every order, including $W_2\\le 7q^{-1/3}$ and $W_\\infty\\ll q^{-1/6}$. These estimates follow from a general theorem for probability measures satisfying Katz-type Chebyshev moment bounds. Its proof combines an edge-sensitive discrepancy estimate with a new level-set bound for the distribution function and a quantile-displacement argument. We construct measures showing that the resulting exponents cannot in general be improved using moment bounds of this type alone, in the trace metric and in the angle metric for $s\\le 2$ and $s\\ge 6$ (up to logarithms); these constructions do not assert optimality for the Kloosterman measures themselves. A sharp comparison between Wasserstein distances and the Kolmogorov distance for the Sato-Tate law exhibits the same transition at $s=3$. Finally, using the exact third moment, we prove $W_1(μ_q,μ_{\\mathrm{ST}})>q^{-1/2}/10$ when $3\\nmid q$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16169",
  "author_total": 1,
  "authors": [
    {
      "name": "Amir M. Rahimi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fppqw0s77d6hs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16169",
  "fulltext_url": null,
  "id": "fppqw0s77d6hs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16169",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221787230",
  "pdf_url": "https://arxiv.org/pdf/2610.16169",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fppqw0s77d6hs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16169",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16169"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Wasserstein rates of all orders for the Sato-Tate law of Kloosterman sums",
  "title_text": "Wasserstein rates of all orders for the Sato-Tate law of Kloosterman sums",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:38Z",
  "url": "https://research.draconitic.com/paper/fppqw0s77d6hs",
  "venue": "arXiv",
  "versions": []
}