When Can Patterns Characterise Fractional Powers? On a Question of Shallit

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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-4425-012 (Shallit, On patterns and pattern avoidance, Oberwolfach Reports 7 (2010), Problem 38).