{
  "abstract": "We address the inverse problem for complexity functions of infinite words, which asks which functions can occur as complexity functions of infinite words. We survey our recent advances on this problem and several of its ramifications. We describe an asymptotic solution to the inverse problem, realizing every function satisfying two necessary conditions as the complexity function of an infinite word up to asymptotic equivalence. We then turn to phenomena beyond asymptotic equivalence, presenting constructions of infinite words with seemingly contradictory behavior of the discrete derivatives of their complexity functions. These lead to a solution of a 1995 problem of Cassaigne and to applications in symbolic dynamics, including the question of convergence of Rauzy graphs, as well as to the growth and asymptotic invariants of algebraic structures.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics and Statistics, Hunter College, City University of New York, New York, NY 10065",
    "Department of Pure Mathematics, Xi’an Jiaotong–Liverpool University, Suzhou, Jiangsu 215123, China",
    "Southern University of Science and Technology International Center for Mathematics, Shenzhen, Guangdong 518055, China",
    "Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, RJ 22460-320, Brazil"
  ],
  "article_type": "research article",
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Be’eri Greenfeld"
    },
    {
      "affs": [
        1
      ],
      "name": "Paul-Henry Leemann"
    },
    {
      "affs": [
        2,
        3
      ],
      "name": "Carlos Gustavo Moreira"
    },
    {
      "affs": [
        2
      ],
      "name": "Efim Zelmanov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cdqrynvetmx6t.bib",
  "cited_by": 0,
  "doi": "10.1073/pnas.2627766123",
  "fulltext_url": "https://pmc.ncbi.nlm.nih.gov/articles/PMC13732050/",
  "id": "cdqrynvetmx6t",
  "issns": [
    "0027-8424",
    "1091-6490"
  ],
  "landing_url": "https://doi.org/10.1073/pnas.2627766123",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "heading",
      "421"
    ],
    [
      "heading",
      "Mathematics"
    ],
    [
      "heading",
      "Physical Sciences"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7224476970",
  "pdf_url": null,
  "pmcid": "PMC13732050",
  "pmid": "42956034",
  "published": "2026-10-29",
  "publisher": "National Academy of Sciences",
  "ris": "https://research.draconitic.com/paper/cdqrynvetmx6t.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:22:06Z",
      "native_id": "PMC13732050",
      "source": "pmc",
      "url": "https://doi.org/10.1073/pnas.2627766123"
    }
  ],
  "subjects": [
    {
      "field": "biochemistry",
      "origin": "journal-openalex:1312|Genomics and Chromatin Dynamics",
      "subfield": "molecular-biology"
    }
  ],
  "title": "Complexity functions of infinite sequences",
  "title_text": "Complexity functions of infinite sequences",
  "type": "peer-reviewed",
  "updated_at": "2026-11-08T12:22:06Z",
  "url": "https://research.draconitic.com/paper/cdqrynvetmx6t",
  "venue": "Proceedings of the National Academy of Sciences of the United States of America",
  "versions": []
}