{
  "abstract": "A family $\\mathcal F$ of finite subsets of $\\mathbb Z$ is packed into $[N]$ if suitable integer translates of its members are pairwise disjoint subsets of $[N]$. We study two prescribed-difference packing problems of Alon, Dębski, Grytczuk and Przybyło for the arithmetic progressions $A_d=\\{d,2d,\\ldots,\\lfloor n/d\\rfloor d\\}$ and $B_d=\\{d,2d,\\ldots,nd\\}$. Let $m(n)$ and $M(n)$ denote the corresponding minimum packing lengths, with subscripts indicating restrictions on the admissible differences, and let $\\mathcal P(x)$ denote the set of primes at most $x$. The key ingredient is a residue-class selection scheme that encodes pairwise intersection constraints by cyclic intervals. A lattice-covering argument yields a simultaneous admissible choice, permitting substantial overlap of the containing intervals and providing the sharp upper bounds needed for the bounded-diameter family and the prime-difference equal-cardinality family. For the bounded-diameter family, we prove $m(n)\\sim m_{\\mathcal P(\\sqrt n)}(n)\\sim 4n^{3/2}/(3\\log n)$. For the equal-cardinality family with prime differences, we prove $M_{\\mathcal P(n)}(n)\\sim n^3/(6\\log n)$. For the full equal-cardinality family, we show $M(n)\\ge \\left(\\frac{19}{108}-o(1)\\right)n^3/\\log n$. Together with corresponding estimates for restricted ranges of differences, these results prove several conjectures of Alon--Dębski--Grytczuk--Przybyło and disprove others.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07487",
  "author_total": 3,
  "authors": [
    {
      "name": "Jianfeng Hou"
    },
    {
      "name": "Siyue Liu"
    },
    {
      "name": "Hongbin Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ccg4ty1g1kj5z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07487",
  "fulltext_url": null,
  "id": "ccg4ty1g1kj5z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07487",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212019823",
  "pdf_url": "https://arxiv.org/pdf/2609.07487",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ccg4ty1g1kj5z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.07487",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07487"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Simultaneous Residue-Class Selection in Prescribed-Difference Packings",
  "title_text": "Simultaneous Residue-Class Selection in Prescribed-Difference Packings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/ccg4ty1g1kj5z",
  "venue": "arXiv",
  "versions": []
}