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 \(\alpha>1\) that is not an integer, there is a set \(P\) of patterns, finite or infinite, such that a word contains an \(\alpha\)-power, that is, a factor which is an exact \(p/q\)-power with \(p/q\ge\alpha\), 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 \(\alpha>1\). 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 \(\alpha<2\); over \(s\ge3\) letters it exists if and only if \(\alpha\le 1+1/s\), for every non-integer \(\alpha>1\) outside the window \(2\)\({}-2^{-16}\)\({}<\alpha\)\({}<2\), which remains open; over one letter it always exists. Such a set exists if and only if the words of critical exponent at least \(\alpha\) are closed under nonerasing morphisms, so a negative answer is certified by a word \(x\) and a nonerasing morphism \(\sigma\) with \(\operatorname{ce}(x)\)\({}\ge\alpha\)\({}>\operatorname{ce}(\sigma(x))\). The negative answer in the universal reading is a short elementary argument, possibly folklore, in which the alphabet grows as \(\alpha\) approaches an integer; the source reports that Currie and Cassaigne suggested ways to show impossibility for restricted ranges of \(\alpha\), 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 \(5\)\(/3\)\({}<\alpha\)\({}\le2\)\({}-2^{-16}\)) 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.