{
  "abstract": "We introduce and study $\\mathbb{S}$-rich fixed-degree addition chains of a fixed index $n>1$: increasing sequences of positive integers beginning at $1$ in which every term is the sum of at most $d$ earlier terms and which contain all elements of the set $\\mathbb{S}\\cap [1,n]$ as targets. Our main contribution is a simple constructive upper bound for the minimal length $\\ell_n^d(\\mathbb{S})$ of such a chain. Writing the ordered elements of $\\mathbb{S} \\cap [1,n]$ as $z_1\\leq z_2\\leq \\cdots$ and letting $\\nu_d(\\cdot)$ denote the sum of base-$d$ digits, we show $$ \\ell_{n}^d(\\mathbb{S})\\leq \\left \\lfloor\\frac{\\log n}{\\log d}\\right \\rfloor+\\left \\lceil \\frac{\\nu_d(z_1)}{d-1}\\right \\rceil+\\sum \\limits_{i=2}^{|\\mathbb{S}_n|}\\left \\lceil \\frac{\\nu_d(z_i-z_{i-1})}{d-1} \\right \\rceil. $$ a bound that generalizes the standard base-expansion construction for single targets and mirrors the digit-sum phenomenon appearing in known lower bounds. The proof is constructive and yields a straightforward algorithm for chain construction; we record corollaries for prime-rich and square-free sets, give illustrative examples, and discuss tightness, and natural open problems about optimal multi-target precomputation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Theophilus Agama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/50r3m2jq5jymw.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "50r3m2jq5jymw",
  "issns": [],
  "landing_url": "https://hal.science/hal-05756220v1",
  "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-05756220/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/50r3m2jq5jymw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:36Z",
      "native_id": "hal-05756220",
      "source": "hal",
      "url": "https://hal.science/hal-05756220v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "MINIMAL LENGTH OF FIXED DEGREE S-RICH CHAINS",
  "title_text": "MINIMAL LENGTH OF FIXED DEGREE S-RICH CHAINS",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:36Z",
  "url": "https://research.draconitic.com/paper/50r3m2jq5jymw",
  "venue": "HAL",
  "versions": []
}