The monadic theory of toric words

For which unary predicates $P_1,\dots ,P_m$ is the MSO theory of the structure $〈N;<,P_1,\dots ,P_m〉$ decidable? We survey the state of the art, leading us to investigate combinatorial properties of almost-periodic, morphic, and toric words. In doing so, we show that if each $P_i$ can be generated by a toric dynamical system of a certain kind, then the attendant MSO theory is decidable. We give various applications of toric words, including the recent result of [1] that the MSO theory of $〈N;<,\{2^n:n\in N\},\{3^n:n\in N\}〉$ is decidable.