{
  "abstract": "We prove a quantitative Halász theorem for multiplicative functions on the nonzero ideals of $\\mathbb{Z}[i]$, with bounds controlled by pretentious distance to the Archimedean characters $N^{it}$. We also prove a sectorial analogue: under angular non-pretentiousness, the sum of $f$ over ideals lying in a fixed sector is asymptotically given by the expected proportion of the unrestricted sum. Finally, under angular non-pretentiousness and a non-degeneracy condition on conjugate prime pairs, we prove a sectorial short-interval version of the Halász theorem for annular sectors whose radial thickness tends to infinity. The proof of the sectorial short-interval Halász theorem uses angular Fourier expansion, norm-compression to multiplicative functions on $\\mathbb{N}$, and a theorem of Mangerel.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.19105",
  "author_total": 1,
  "authors": [
    {
      "name": "Jan Kuś"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/faarapkpdkvcj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.19105",
  "fulltext_url": null,
  "id": "faarapkpdkvcj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.19105",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7161820778",
  "pdf_url": "https://arxiv.org/pdf/2605.19105",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/faarapkpdkvcj.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:28Z",
      "native_id": "2605.19105",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.19105"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Halász theorems for Gaussian ideals in sectors and short intervals",
  "title_text": "Halász theorems for Gaussian ideals in sectors and short intervals",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:28Z",
  "url": "https://research.draconitic.com/paper/faarapkpdkvcj",
  "venue": "arXiv",
  "versions": []
}