{
  "abstract": "We study multiplicative structures arising from \\emph{fixed-degree} addition chains: increasing sequences $$ s_0=1<s_1<\\cdots<s_h $$ in which each term is the sum of at most $d\\geq 2$ earlier terms (repetition allowed). Writing $\\mathcal{P}(L)$ for the set of integers that can be expressed as a product of distinct terms drawn from a (finite or infinite) fixed-degree chain $L=\\{T_m\\}$, we develop a structural partition of chain steps into four classes $\\mathcal{A}_d,\\mathcal{B}_d,\\mathcal{C}_d,\\mathcal{D}_d$ (dilate / large / medium / small) and introduce the notion of a \\emph{run} of a given step type. By deriving precise two-sided log-sum (product) estimates along pure and mixed-type runs and translating these into elementary counting bounds (for example, run-level bounds of the shape $\\ll(\\log x)^{1/2}$ in the logarithmic scale), we obtain a global sparsity result: under the mild structural hypothesis that the number of runs of types $\\mathcal{D}_d$ and $\\mathcal{A}_d\\cup\\mathcal{D}_d$ on initial segments of the chain is $O_d(1)$, the product-set $\\mathcal{P}(L)$ has zero asymptotic density. In particular, for any such infinite chain $L$ one has $$ \\lim_{x\\longrightarrow \\infty}\\frac{\\#\\{v\\leq x~:~v\\in\\mathcal{P}(L)\\}}{x}=0. $$ The proof combines combinatorial run-analysis, elementary exponential growth estimates along runs, and an assembly of run-level contributions into explicit global counting bounds; quantitative run-level inequalities (involving the clearly defined parameters $\\lambda_d$ and a slowly varying $\\delta(m)=1/\\log m$) are recorded and may be of independent interest for further study of fixed-degree chains",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Theophilus Agama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/byp24829k9kn6.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "byp24829k9kn6",
  "issns": [],
  "landing_url": "https://hal.science/hal-05784851v1",
  "language": "en",
  "licence": "CC-BY-ND-4.0",
  "licence_label": "CC BY-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05784851/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/byp24829k9kn6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:50Z",
      "native_id": "hal-05784851",
      "source": "hal",
      "url": "https://hal.science/hal-05784851v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "PRODUCT DISTRIBUTION IN A FIXED DEGREE CHAIN",
  "title_text": "PRODUCT DISTRIBUTION IN A FIXED DEGREE CHAIN",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:50Z",
  "url": "https://research.draconitic.com/paper/byp24829k9kn6",
  "venue": "HAL",
  "versions": []
}