Fixed-alphabet bounds for powers under injective morphisms

Determine whether, for a fixed alphabet of ternary or larger size, primitive words in which every letter occurs a number of times different from one can be mapped to powers of unbounded exponent by injective morphisms, or whether there is a constant bound such that every such primitive word either maps only to powers up to that bound or maps to powers of arbitrarily high exponent.

Background

The paper characterizes when a word can be mapped to arbitrarily high powers by injective morphisms: this occurs precisely when some letter occurs exactly once in the word’s primitive root. For all other primitive words, the paper proves that the largest attainable exponent is strictly smaller than the word length.

When the alphabet size is allowed to grow, the paper constructs examples showing that the largest attainable exponent can grow linearly with word length. For binary alphabets, the paper proves a constant bound, with the only unbounded cases arising from the previously characterized trivial situation involving a letter occurring exactly once. The unresolved issue is whether analogous uniform bounds hold for every fixed ternary or larger alphabet, or whether arbitrarily large exponents can occur without any letter having frequency one.