{
  "abstract": "Given a real $x\\in 2^ω$, how many reals are there that are simpler (or equally simple) to describe than $x$. With ``simpler to describe\" characterized by $K$-complexity of initial segments (hence we call such $y$ with $y\\leq_K x$ a solution to $x$), if $x$ is infinitely often $K$-trivial (hence i.o. $K$-trivial), then a simple counting argument shows that the number of solutions to $x$ is countable. If $x$ is non i.o. $K$-trivial, then must it admit uncountably many solutions? Motivated by this question, we show that: (1) There is a non i.o. $K$-trivial $x$ such that most of its solution is i.o. $K$-trivial. i.e., Only countably many solution to $x$ are non i.o. $K$-trivial. In particular, the cardinality of $K$-degree of $x$ is countable. (2) There is a minimal pair of K-degree (meaning their common solutions are all $K$-trivial) so that each of the pair is non i.o. $K$-trivial, generalizing a result of Csima and Montalbán (where the pair is not required to be non i.o. $K$-trivial but non $K$-trivial). (3) There is a non i.o. $K$-trivial $x$ whose $Δ_K$ complexity grows arbitrarily slow (in a weak sense: $\\inf_{m\\geq n}Δ_K(x\\upharpoonright m)\\leq h(n)$ with $h$ a given unbounded and nondecreasing function) such that $x$ admits uncountably many solutions. (4) An ample excess lemma holds for non i.o. $K$-trivial. Namely: $$ \\sum_n 2^{-(K(x\\upharpoonright n)- \\log_2 n)}<\\infty. $$ for all non i.o. $K$-trivial $x$. Actually, this even holds for non i.o. $b$-$K$-trivial with $b$ only depends on the universal prefix free machine. The same argument also gives a constant complexity return phenomenon for $K$-trivial where this constant only depends on the universal prefix free machine.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39172",
  "author_total": 2,
  "authors": [
    {
      "name": "George Barmpalias"
    },
    {
      "name": "Lu Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/22sf3mjn5xerj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39172",
  "fulltext_url": null,
  "id": "22sf3mjn5xerj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39172",
  "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.LO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39172",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/22sf3mjn5xerj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39172",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39172"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Cardinality of the K-cones below a non infinitely often $K$-trivial",
  "title_text": "Cardinality of the K-cones below a non infinitely often K-trivial",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/22sf3mjn5xerj",
  "venue": "arXiv",
  "versions": []
}