{
  "abstract": "Majorization is a fundamental tool for comparing how \"spread out\" the entries of two vectors are. Key majorization results were obtained for the integers by Hardy, Littlewood and Pólya and for $\\mathbb{Z}_p$ by Lev. In this paper, we establish a powerful majorization theorem in $\\mathbb{Z}_2^n$ that is an analogue of Lev's result in $\\mathbb{Z}_p$. Our proof is based on a novel use of a compression argument that optimizes certain sums of additive representation counts. Our majorization theorem has several applications in additive combinatorics. We resolve the question: given subsets $A_1,\\ldots,A_k\\subseteq\\mathbb{Z}_2^n$ of prescribed sizes, when is the function $r_k(A_1,\\ldots,A_k)$, which counts the number of additive $k$-tuples in $A_1\\times\\cdots\\times A_k$, maximized? We establish the corresponding minimization result and a characterization of all extremizers for $r_k(A,\\ldots,A)$ when $k$ is odd. When $k=3$, this quantity is the number of Schur triples in $A$; as a special case, we recover a theorem of Samotij and Sudakov on Schur triples. We also obtain a convexity inequality and use it to prove an analogue of Pollard's theorem that strengthens and extends a well-known result of Bollobás and Leader in the $\\mathbb{Z}_2^n$ setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07855",
  "author_total": 3,
  "authors": [
    {
      "name": "Sophie Huczynska"
    },
    {
      "name": "Firdavs Rakhmonov"
    },
    {
      "name": "Chi Hoi Yip"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4v934vnsvbhze.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07855",
  "fulltext_url": null,
  "id": "4v934vnsvbhze",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07855",
  "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.CO"
    ],
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      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
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  "openalex": "W7212078026",
  "pdf_url": "https://arxiv.org/pdf/2609.07855",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4v934vnsvbhze.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07855",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07855"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Majorization and additive tuples in $\\mathbb{Z}_2^n$",
  "title_text": "Majorization and additive tuples in ℤ_2^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/4v934vnsvbhze",
  "venue": "arXiv",
  "versions": []
}