An open conjecture of Hartmannis and Stearns is that a real number where the sequence of digits is produced by a linear time Turing machine is either rational or transcendental. This is considered a hard problem, and implies, among other things, that integer multiplication cannot be done in linear time. Much work has been done on weaker forms of this conjecture, pulling in tools such as the Mahler Method and the Subspace Theorem. In this talk I will discuss some results concerning the transcendence of (Epi)Sturimian words, which include the d-bonacci numbers, which will be our prototypical example. We do this by introducing a combinatorial criterion on words, called echoing, which implies transcendence. I will then give some further applications of this criterion. This is joint work with Pavol Kebis, Florian Luca, Joel Ouaknine, and James Worrell.
