{
  "abstract": "In this paper, we investigate whether Harper's better than square-root cancellation phenomenon still appears in Piatetski-Shapiro sequences, and when one should expect only square-root size. If $c=c(X)$ satisfies $(c-1)\\log X\\to 0$ as $X\\to\\infty$, then for Steinhaus or Rademacher random multiplicative functions we have \\[ \\mathbb{E}\\left|\\sum_{n\\in\\mathcal{N}^{(c)}(X)}f(n)\\right|=o\\!\\left(\\sqrt{|\\mathcal{N}^{(c)}(X)|}\\right). \\] For Lebesgue almost every fixed $1<c<\\frac{8}{7}$, we prove the multiplicative energy satisfies \\[ E^\\times(\\mathcal{N}^{(c)}(X))=2|\\mathcal{N}^{(c)}(X)|^2-|\\mathcal{N}^{(c)}(X)|+o\\left(|\\mathcal{N}^{(c)}(X)|^2\\right). \\] Together with a largest prime factor estimate, this allows us to apply the criterion of Soundararajan and Xu and obtain a standard complex Gaussian limit for the normalized Steinhaus sum. To the best of our knowledge, this gives the first asymptotic formula for the multiplicative energy of Piatetski-Shapiro sequences. For fixed exponent argument, our proof combines metric arguments with the Van der Corput's $B$-process and a double large sieve estimate with Robert-Sargos spacing theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15807",
  "author_total": 2,
  "authors": [
    {
      "name": "Renjie Zhu"
    },
    {
      "name": "Tianping Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ektwy0yabpj1k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15807",
  "fulltext_url": null,
  "id": "ektwy0yabpj1k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15807",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7203710312",
  "pdf_url": "https://arxiv.org/pdf/2608.15807",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ektwy0yabpj1k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2608.15807",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15807"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Better than square-root cancellation and Gaussian behavior on Piatetski-Shapiro Sequences",
  "title_text": "Better than square-root cancellation and Gaussian behavior on Piatetski-Shapiro Sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/ektwy0yabpj1k",
  "venue": "arXiv",
  "versions": []
}