{
  "abstract": "Let \\[ D(f,h)=\\sum_{k=1}^n a_k f(b_kh) \\] be a finite-difference operator, where the coefficients may be complex and the nodes are real. We study the asymptotic implication \\[ D(f,h)=o(h)\\quad\\Longrightarrow\\quad f(h)=o(h) \\] for Lebesgue measurable functions. In 2017, Ash, Catoiu, and Csörnyei \\cite{ACC} showed that this implication holds for every measurable function precisely for operators belonging to a very restricted class of finite differences. To establish that every operator outside the exceptional class admits a counterexample, Ash, Catoiu, and Csörnyei analyze an infinite system of linear equations. Their argument is therefore existential in nature and does not provide an explicit description of the counterexamples. By contrast, we give a substantially simpler and elementary construction based on zeros of finite Laurent polynomials and multiplicative characters on finitely generated multiplicative subgroups. This provides a direct recipe for constructing measurable functions \\(f\\) such that \\[ D(f,h)=0\\qquad\\text{for every }h\\in\\mathbb C, \\] while, in fact, \\[ f(h)\\not=o(1)\\qquad\\text{as }h\\to0. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30583",
  "author_total": 1,
  "authors": [
    {
      "name": "Hajrudin Fejzić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0mge4ydfb86bw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30583",
  "fulltext_url": null,
  "id": "0mge4ydfb86bw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30583",
  "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.CA"
    ],
    [
      "openalex",
      "2610|Mathematical and Theoretical Analysis"
    ]
  ],
  "openalex": "W7223363114",
  "pdf_url": "https://arxiv.org/pdf/2610.30583",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0mge4ydfb86bw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.30583",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30583"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Explicit Measurable Counterexamples to Finite-Difference Operators",
  "title_text": "Explicit Measurable Counterexamples to Finite-Difference Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/0mge4ydfb86bw",
  "venue": "arXiv",
  "versions": []
}