{
  "abstract": "Building on the large sieve, we prove a new sieve estimate for sets which are missing various residue classes modulo prime squares. For certain interesting residue class configurations, our bound is a significant improvement over the corresponding Johnsen-Selberg sieve inequality. Our application of this sieve estimate is motivated by Ostmann's problem. We study additive decompositions of sums of squares and show that any non-trivial sumset decomposition must consist of two sets of roughly equal size: If $A_1, A_2 \\subseteq \\mathbb{Z}$ with $A_1 + A_2 = \\{x^2 + y^2 : x, y \\in \\mathbb{Z}\\}$ and $|A_1|, |A_2| \\geq 2$, then $\\sqrt{x}/(\\log x)^{9/2} \\ll |A_i \\cap [1, x]| \\ll \\sqrt{x}(\\log x)^4$ for $i \\in \\{1, 2\\}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20554",
  "author_total": 2,
  "authors": [
    {
      "name": "Christian Elsholtz"
    },
    {
      "name": "Julia Stadlmann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b43ks29stj3ca.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20554",
  "fulltext_url": null,
  "id": "b43ks29stj3ca",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20554",
  "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": "W7221887076",
  "pdf_url": "https://arxiv.org/pdf/2610.20554",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b43ks29stj3ca.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20554",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20554"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The large sieve inequality and additive decompositions of sums of squares",
  "title_text": "The large sieve inequality and additive decompositions of sums of squares",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/b43ks29stj3ca",
  "venue": "arXiv",
  "versions": []
}