{
  "abstract": "A set $B$ in an abelian group is a $B_h$ set if every $h$-term sum has a unique representation up to permutation; for $h=2$ these are the Sidon sets. We study a weighted removal problem for this collision-free property: if the $h$-fold sum map has small Rényi entropy loss, how much probability mass must be deleted so that the remaining support is a $B_h$ set? Two Rényi orders arise: $α$ is the order at which the coarsening loss is measured, whereas $β$ is the order of the entropy constraint. The diagonal specialization $β=α$ ties the two roles together. We determine the resulting stability problem on the positive $(α,β)$-quadrant. Stability holds exactly when $β\\le1$ and $α\\geβ$. Inside this region the optimal deletion rate is polynomial for $β<1$ and logarithmic on the boundary $β=1$, where the leading constant is exact; outside it, stability fails through two distinct mechanisms: a supercritical budget and dilution by light atoms. In both unstable regimes the limiting defect is computed exactly. The upper bounds follow from a coarsening inequality with best possible constant, which also yields an entropy-free removal theorem, a finite combinatorial consequence for moments of the representation function, and extensions to $B_h[g]$ sets. Matching constructions show that the phase boundaries and rates are sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23922",
  "author_total": 1,
  "authors": [
    {
      "name": "Jae Oh Woo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/42w1tkb3hqsn1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23922",
  "fulltext_url": null,
  "id": "42w1tkb3hqsn1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23922",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214019417",
  "pdf_url": "https://arxiv.org/pdf/2609.23922",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/42w1tkb3hqsn1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.23922",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23922"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Rényi stability of $B_h$ sets: a two-order phase diagram and sharp deletion principles",
  "title_text": "Rényi stability of B_h sets: a two-order phase diagram and sharp deletion principles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/42w1tkb3hqsn1",
  "venue": "arXiv",
  "versions": []
}