Repetition Threshold for Binary Automatic Sequences

TL;DR

This paper demonstrates that for any integer k ≥ 2, there exists a binary k-automatic sequence with a critical exponent at most 7/3, extending to Fibonacci-automatic and Tribonacci-automatic sequences.

Contribution

It establishes the existence of binary k-automatic sequences with bounded critical exponent, including Fibonacci and Tribonacci automatic sequences, for all integers k ≥ 2.

Findings

Existence of binary k-automatic sequences with critical exponent ≤ 7/3 for all k ≥ 2

Extension of results to Fibonacci-automatic and Tribonacci-automatic sequences

Provides bounds on the critical exponent for these classes of 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.