Morphic and Automatic Words: Maximal Blocks and Diophantine Approximation

Let $\text{mb}w$ be a morphic word over a finite alphabet $\mathrm{\Sigma }$, and let $\mathrm{\Delta }$ be a nonempty subset of $\mathrm{\Sigma }$. We study the behavior of maximal blocks consisting only of letters from $\mathrm{\Delta }$ in $\text{mb}w$, and prove the following: let $(i_k,j_k)$ denote the starting and ending positions, respectively, of the $k$’th maximal $\mathrm{\Delta }$-block in $\text{mb}w$. Then ${\mathrm{lim\hspace{0.17em}sup}}_{k\to \mathrm{\infty }}(j_k\mathrm{/}{i}_k)$ is algebraic if $\text{mb}w$ is morphic, and rational if $\text{mb}w$ is automatic. As a result, we show that the same conclusion holds if $(i_k,j_k)$ are the starting and ending positions of the $k$’th maximal zero block, and, more generally, of the $k$’th maximal $x$-block, where $x$ is an arbitrary word. This enables us to draw conclusions about the irrationality exponent of automatic and morphic numbers. In particular, we show that the irrationality exponent of automatic (resp., morphic) numbers belonging to a certain class that we define is rational (resp., algebraic).