Quasiperiods, Subword Complexity and the Smallest Pisot Number

A quasiperiod of a finite or infinite string/word is a word whose occurrences cover every part of the string. A word or an infinite string is referred to as quasiperiodic if it has a quasiperiod. It is obvious that a quasiperiodic infinite string cannot have every word as a subword (factor). Therefore, the question arises how large the set of subwords of a quasiperiodic infinite string can be [8]. Here we show that on the one hand the maximal subword complexity of quasiperiodic infinite strings and on the other hand the size of the sets of maximally complex quasiperiodic infinite strings both are intimately related to the smallest Pisot number tP (also known as plastic constant). We provide an exact estimate on the maximal subword complexity for quasiperiodic infinite words.