Minimal unique expansions with digits in ternary alphabets

The study of the redundancy of non-integer base numeration systems involves several fields of mathematics and of theoretical computer science, including number theory, ergodic theory, topology, and combinatorics on words. When the base is smaller than a sharp value, called critical base, only trivial expansions in a non-integer base are unique, while for greater bases there exist some non-trivial unique expansions. By investigating an unexpected relation between balanced sequences and unique expansions, we explicitly characterize for a large class of three-letter alphabets the minimal unique expansions, namely those unique expansions that first appear when we choose bases larger than the critical base.