{
  "abstract": "We prove a refined quadratic normal approximation for the first absolute moment of normalized weighted Rademacher sums with bounded maximal coefficients. For any weight vector $w\\in\\mathbb{R}^{n}$ satisfying $\\|w\\|_{2}=1$ and $\\|w\\|_{\\infty}\\leqβ$ with sufficiently small $β>0$, we establish the uniform error bound $|\\mathbb{E}|\\sum_{i=1}^{n}w_{i}X_{i}|-\\sqrt{2/π}|=O(β^{2})$ over all admissible weight configurations. Our proof combines zero-bias Stein's method and refined small-ball probability estimates to exploit symmetry cancellation and control the non-smooth residual of the absolute-value test function. An explicit extremal construction further verifies the optimality of this quadratic convergence rate. As an application, we establish an asymptotically sharp refinement of the Friedgut--Kalai--Naor (FKN) theorem for Boolean functions, also known as the sharp $π/2$ theorem, characterizing the level-1 Fourier energy for functions deviating far from dictatorships.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29703",
  "author_total": 1,
  "authors": [
    {
      "name": "Lei Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2jt7pd92z28qr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29703",
  "fulltext_url": null,
  "id": "2jt7pd92z28qr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29703",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7204922522",
  "pdf_url": "https://arxiv.org/pdf/2608.29703",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2jt7pd92z28qr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29703",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29703"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp $π/2$ Theorem",
  "title_text": "A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp π/2 Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/2jt7pd92z28qr",
  "venue": "arXiv",
  "versions": []
}