{
  "abstract": "We prove a sharp lower bound for the cardinality of sumsets of subsets of $\\mathbb{Z}^d$ confined to a hypercube, resolving in strong form a conjecture that was made explicit by Becker, Ivanisvili, Krachun and Madrid and had circulated in the folklore of the field for some time. Specifically, for sets $A_j\\subseteq \\{0,1,2,\\dots,m\\}^d$ we show that $$|A_1+\\dots+A_n|\\;\\geq\\; (|A_1|\\cdots|A_n|)^{1/p},\\qquad p=\\frac{n\\log(m+1)}{\\log(nm+1)},$$ with the exponent best possible. The only previously known sharp cases were $A_j\\subseteq \\{0,1\\}^d$, for all $n\\ge1$, and $A_j\\subseteq \\{0,1,2\\}^d$ for $n=2$. We also prove a sharp inequality in the case when $A_j\\subseteq\\{0,1,\\dots,m_j\\}^d$ for different $m_j$. We obtain the above inequality as a corollary of a stronger result on sup-convolution of functions on $\\mathbb{Z}^d$, whose proof is based on a novel mixed volume representation of a lattice path norm, together with a sharp one-dimensional functional inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Felipe Gonçalves"
    },
    {
      "name": "Danylo Radchenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3sj9k3ghspa9y.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "3sj9k3ghspa9y",
  "issns": [],
  "landing_url": "https://hal.science/hal-05851856v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-ca"
    ],
    [
      "hal",
      "1.math.math-co"
    ],
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      "hal",
      "1.math.math-mg"
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  "openalex": null,
  "pdf_url": "https://hal.science/hal-05851856/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3sj9k3ghspa9y.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:13Z",
      "native_id": "hal-05851856",
      "source": "hal",
      "url": "https://hal.science/hal-05851856v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-ca",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-co",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-mg",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Sharp lower bounds for sumsets in hypercubes",
  "title_text": "Sharp lower bounds for sumsets in hypercubes",
  "type": "preprint",
  "updated_at": "2026-11-06T12:20:44Z",
  "url": "https://research.draconitic.com/paper/3sj9k3ghspa9y",
  "venue": "HAL",
  "versions": []
}