Dense Languages and Non Primitive Words

In this paper, we are concerned with dense languages and non primitive words. A language L is said to be dense if any string can be found as a substring of element of L. In 2020, Ryoma Syn'ya proved that any regular language with positive asymptotic density always containsinfinitely many non-primitive words. Since positive asymptotic density implies density, it is natural to ask whether his result can be generalized for a wider class of dense languages. In this paper, we actually obtain such generalization.