The Tribonacci constant and finite automata

Abstract

We show that there is no automaton accepting the Tribonacci representations of $n$ and $x$ in parallel, where $\ddot{I}ˆ=1.839\cdots$ is the Tribonacci constant, and $x=\lfloor n\ddot{I}ˆ\rfloor$. Similarly, there is no Tribonacci automaton generating the Sturmian characteristic word with slope $\ddot{I}ˆ-1$.