On the boundary sequence of an automatic sequence (Q2237216)

A `boundary' of a word \(w_i\cdots w_{i+n}\) over a finite alphabet is the word \(w_iw_{i+n}\), and the set of all boundaries of words of length \(n+1\) appearing in a sequence \(w=w_1w_2\cdots\) is denoted \(\partial_{w}(n)\), giving a `boundary sequence' \((\partial_w(n))_{n\ge 1}\) associated to \(w\). \textit{J. Chen} and \textit{Z.-X. Wen} [Theor. Comput. Sci. 780, 66--73 (2019; Zbl 1423.68369)] computed the boundary sequence of the generalized Thue-Morse sequence, and conjectured that the boundary sequence of a \(b\)-automatic sequence is \(b\)-automatic. This is proved here, and the boundary sequence of generalized Cantor sequences is found. Periodic boundary sequences are characterized, and these results are applied to show that certain abelian complexities are automatic.