{
  "abstract": "A zero set for a space of analytic functions is a set on which some nonzero function in the space vanishes; it is exact if it coincides with the full zero set of such a function. We construct zero sets that are not exact in the classical coefficient spaces $\\ell_A^p$ of functions analytic on the unit disk for every $1\\le p<2$. For $1<p<2$, a single Blaschke set $W$ is a zero set for every $\\ell_A^p$ in this range, but is not exact for any of them. The endpoint case $p=1$, corresponding to the analytic Wiener algebra $\\ell_A^1$, requires a separate construction, yielding a non-exact zero set that accumulates at only one point of the unit circle. The proofs combine quadratic characters, finite-dimensional Clark transforms, and logarithmic estimates in zero-free regions. We also establish sharp moment duality and an exact summability criterion for a family of regular polygons. For $p>2$, we construct two disjoint exact zero sets whose union is a uniqueness set. For these exponents, suitably chosen rotations also yield a continuum-sized family of nonzero closed shift-invariant subspaces of $\\ell^p$ with pairwise trivial intersections, recovering a known existence result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02159",
  "author_total": 3,
  "authors": [
    {
      "name": "Evgeny Abakumov"
    },
    {
      "name": "Isabelle Chalendar"
    },
    {
      "name": "Alexei Poltoratski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8a66bgdp21pzp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02159",
  "fulltext_url": null,
  "id": "8a66bgdp21pzp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02159",
  "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.CV"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02159",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8a66bgdp21pzp.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02159",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02159"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Zero sets in spaces of analytic functions with $p$-summable Taylor coefficients",
  "title_text": "Zero sets in spaces of analytic functions with p-summable Taylor coefficients",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/8a66bgdp21pzp",
  "venue": "arXiv",
  "versions": []
}