Subshifts with Slowly Growing Numbers of Follower Sets

For any subshift, define $F_X(n)$ to be the collection of distinct follower sets of words of length $n$ in $X$. Based on a similar result of the second and third authors, we conjecture that if there exists an $n$ for which $|F_X(n)| \leq n$,then$X$ is sofic. In this paper, we prove several results related to this conjecture, including verifying it for $n\le 3$, proving that the conjecture is true for a large class of coded subshifts, and showing that if there exists $n$ for which $\mathrm{\mid }{F}_X(n)\mathrm{\mid }\le {\log }_2(n+1)$, then $X$ is sofic.