{
  "abstract": "Given a finite subset $A\\subseteq\\mathbb{Z}$ and a multiset of non-zero integers $Λ=\\{λ_1, \\ldots, λ_k\\}$, the dilate sum \\[Λ\\cdot A:=λ_1\\cdot A+λ_2\\cdot A+\\cdots+λ_k\\cdot A =\\{λ_1a_1+\\cdots+λ_ka_k: a_1, \\ldots, a_k\\in A\\}.\\] For a non-zero real number $r$, say that $Λ_2$ $r$-dominates $Λ_1$ if we can partition some of the elements of $Λ_2$ into blocks summing to the elements of $r\\cdotΛ_1$. Then for all finite subsets $A\\subseteq\\mathbb{Z}$, $Λ_2\\cdot A$ contains a translate of $r\\cdot(Λ_1\\cdot A)$, and hence $|Λ_1\\cdot A|\\le|Λ_2\\cdot A|$ for all $A$. In a recent paper, Kravitz (arXiv:2606.27087) proves the converse: if $Λ_2$ does not $r$-dominate $Λ_1$ for any $r$, then there exists a finite set $A$ satisfying $|Λ_1\\cdot A|>|Λ_2\\cdot A|$. For example, $\\{2, 3, 5, 7\\}$ does not contain two disjoint subsets with a sum ratio of 7, and so there exists a finite $A\\subseteq\\mathbb{Z}$ for which $|A+7\\cdot A|>|2\\cdot A+3\\cdot A+5\\cdot A+7\\cdot A|$. We extend Kravitz's method to apply it to multiple inequalities and multiple sets, showing that for multisets $Λ_{i,j}$ with $1\\le i\\le k$ and $1\\le j\\le\\ell$, there exist finite sets $A_1, \\ldots, A_\\ell\\subseteq\\mathbb{Z}$ such that for all $1\\le i<k$ \\[|Λ_{i,1}\\cdot A_1+\\cdots+Λ_{i,\\ell}\\cdot A_\\ell| >|Λ_{i+1,1}\\cdot A_1+\\cdots+Λ_{i+1,\\ell}\\cdot A_\\ell|\\] provided there do not exist indices $1\\le i_1<i_2\\le k$ and a non-zero scalar $r$ such that $Λ_{i_2,j}$ $r$-dominates $Λ_{i_1,j}$ for all $1\\le j\\le\\ell$. Along the way, we resolve the analogue of Nathanson's Problem 1 for his Theorem 4 (arXiv:1604.08940), as well as the problems in Section 1.4.3 of Kravitz's paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00217",
  "author_total": 1,
  "authors": [
    {
      "name": "Luke Pebody"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bgewgjmqyv3dw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00217",
  "fulltext_url": null,
  "id": "bgewgjmqyv3dw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00217",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00217",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bgewgjmqyv3dw.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00217",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00217"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Dilate Rollercoaster",
  "title_text": "The Dilate Rollercoaster",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/bgewgjmqyv3dw",
  "venue": "arXiv",
  "versions": []
}