Repetition Threshold for Binary Automatic Sequences

Abstract

The critical exponent of an infinite word $\bf x$ is the supremum, over all finite nonempty factors $f$, of the exponent of $f$. In this note we show that for all integers $k\geq 2,$ there is a binary infinite $k$-automatic sequence with critical exponent $\leq 7/3$. The same conclusion holds for Fibonacci-automatic and Tribonacci-automatic sequences.