{
  "abstract": "For real numbers $α$ and $β$, let $\\mathcal{B}(α,β)=\\{\\lfloorαm+β\\rfloor:m\\in\\mathbb{N}\\}$. Wooley asked whether the exponential sum over $\\mathcal{B}(α,β)\\cap[1,N]$ always has $L^1$ norm $O_{α,β}(\\log N)$. We give a negative answer. We construct fixed $α>1$ and $β$, with $1/α$ of finite Diophantine type, such that the $L^1$ norm is $\\gg(\\log N)^2$ for infinitely many $N$. Wooley also proved that this $L^1$ norm is $O_{α,β}((\\log(2N))^2)$ when $1/α$ has finite Diophantine type. Our example shows that his upper bound is sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01494",
  "author_total": 1,
  "authors": [
    {
      "name": "Dijia Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8p00kc63zjfcn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01494",
  "fulltext_url": null,
  "id": "8p00kc63zjfcn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01494",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01494",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8p00kc63zjfcn.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01494",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01494"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Counterexample to Wooley's $L^1$ Question for Beatty Sequences",
  "title_text": "A Counterexample to Wooley's L^1 Question for Beatty Sequences",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/8p00kc63zjfcn",
  "venue": "arXiv",
  "versions": []
}