{
  "schema_version": 1,
  "problem_number": "OWR-4425-012",
  "title": "When Can Patterns Characterise Fractional Powers? On a Question of Shallit",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Shallit asked (Problem 38 in Oberwolfach Reports 7 (2010)) whether, for a real number α > 1 that is not an integer, there is a set P of patterns, finite or infinite, such that a word contains an α-power, that is, a factor which is an exact p/q-power with p/q ≥ α, if and only if some factor of the word matches a pattern of P. If one set P must serve words over every finite alphabet, as in the source's example of overlaps, the answer is negative for every non-integer α > 1. For a fixed alphabet of s letters, where P may depend on the alphabet, we show: over two letters such a set exists if and only if α < 2; over s ≥ 3 letters it exists if and only if α ≤ 1 + 1/s, for every non-integer α > 1 outside the window 2 − 2^(−16) < α < 2, which remains open; over one letter it always exists. Such a set exists if and only if the words of critical exponent at least α are closed under nonerasing morphisms, so a negative answer is certified by a word x and a nonerasing morphism σ with ce(x) ≥ α > ce(σ(x)). The negative answer in the universal reading is a short elementary argument, possibly folklore, in which the alphabet grows as α approaches an integer; the source reports that Currie and Cassaigne suggested ways to show impossibility for restricted ranges of α, and we found no write-up of their arguments. The fixed-alphabet results use the Thue–Morse word and its gap words. Three lemmas on explicit binary words and 29 explicit ternary certificates (covering 5/3 < α ≤ 2 − 2^(−16)) are finite statements decided by exact integer computation, each reproduced by several independent programs; all infinite families are treated by proofs. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "keywords": [
    "combinatorics on words",
    "patterns",
    "nonerasing morphisms",
    "fractional powers",
    "critical exponent",
    "Thue–Morse word",
    "gap words",
    "repetitions",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-4425-012",
    "math.CO",
    "cs.FL",
    "cs.DM",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-4425-012/",
  "pdf_url": "https://eulersolve.org/papers/owr-4425-012/paper.pdf?v=05b7d091d492",
  "doi": "10.5281/zenodo.23486814",
  "zenodo_record_url": "https://zenodo.org/records/23486814",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Negative for every non-integer alpha > 1 if one pattern set must serve all alphabets; for a fixed alphabet, classified except for 2 - 2^-16 < alpha < 2 on three or more letters.",
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    "source.zip": {
      "sha256": "8c9d05947db512fbfc27860b89b740ca4c340a1071d818f78ef4cae7b24c3d21"
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    "verification_report.md": {
      "sha256": "c5d8251130e7a4b05d067932b56c97d2196e9aafc4b3031f3ada8d7091139afe"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}