{
  "abstract": "Let $N=2^k-1$, and let $S_{t,k}$ consist of the pairs $0\\le a,b<N$ such that $a+b\\equiv t\\pmod N$ and $\\wt(a)+\\wt(b)<k$. The Tu--Deng bound $|S_{t,k}|\\le 2^{k-1}$ has recently been proved. In this paper, we give a combinatorial proof of the equality criterion. If the $k$-bit cyclic word of $t$ has $Z$ zeros and $g_1,\\ldots,g_Z$ are the numbers of ones between successive zeros, then $$|S_{t,k}|=2^{k-1} \\quad\\Longleftrightarrow\\quad g_i\\ge Z-1\\quad(1\\le i\\le Z).$$ Moreover, we recover the resulting enumeration of the equality parameters. Beyond equality, if $R\\ge Z\\ge2$, where $R$ is the number of ones, then every nonequality parameter satisfies $$2^{k-1}-|S_{t,k}|\\ge 2^{R-Z+1},$$ and we classify all cases in which this bound is attained. For $R<Z$ we obtain a congruence for $|S_{t,k}|$ and a lower bound for the deficit in terms of the number of cyclic runs of ones.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22451",
  "author_total": 1,
  "authors": [
    {
      "name": "Kaimin Cheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/23hbfy787j5bc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22451",
  "fulltext_url": null,
  "id": "23hbfy787j5bc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22451",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204182798",
  "pdf_url": "https://arxiv.org/pdf/2608.22451",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/23hbfy787j5bc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2608.22451",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22451"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Deficit bounds and equality cases in the Tu--Deng problem",
  "title_text": "Deficit bounds and equality cases in the Tu--Deng problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/23hbfy787j5bc",
  "venue": "arXiv",
  "versions": []
}