{
  "abstract": "The subset $A = \\{a_i:i \\in I\\}$ of a normed vector space is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. Let $\\varepsilon = \\{\\varepsilon_i:i \\in I\\}$ be a set of positive real numbers. An $\\varepsilon$-perturbation of $A$ is a set $A' = \\{a'_i:i\\in I\\}$ such that $|a_i'-a_i|<\\varepsilon_i$ for all $i \\in I$. Let $Δ_{hA} = \\inf\\{|x'-x| : x,x' \\in hA \\text{ and } x\\neq x'\\}$. It is proved that if $A$ is finite or countably infinite set with $Δ_{hA}>0$, then there is a $B_h$-set $A'$ that is an $\\varepsilon$-perturbation of $A$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09249",
  "author_total": 1,
  "authors": [
    {
      "name": "Melvyn B. Nathanson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/88p6xvy962w4z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09249",
  "fulltext_url": null,
  "id": "88p6xvy962w4z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09249",
  "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",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7212176864",
  "pdf_url": "https://arxiv.org/pdf/2609.09249",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/88p6xvy962w4z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09249",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09249"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "$B_h$-sets and perturbations in normed vector spaces",
  "title_text": "B_h-sets and perturbations in normed vector spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/88p6xvy962w4z",
  "venue": "arXiv",
  "versions": []
}