Repetition Threshold for Binary Automatic Sequences

TL;DR

The paper establishes the repetition threshold for binary $k$-automatic sequences, showing $\operatorname{rt}(\mathcal{A}_k)=2$ when $k$ is a power of $2$ and $7/3$ otherwise, with analogous results for Fibonacci-automatic and Tribonacci-automatic sequences. It uses a morphic characterization of $\alpha$-free words, Cobham-type arguments, and explicit $k$-uniform morphisms to construct squarefree or $ (7/3)^+$-free words, aided by the Walnut theorem prover for verification. The Fibonacci-automatic and Tribonacci-automatic cases are handled by building squarefree base words over $\Sigma_4$ and applying a morphism $h$, yielding high-state DFAOs that generate binary sequences with no factors of exponent exceeding $7/3$, and in some cases providing explicit bounds on DFAO sizes. These results highlight that a $7/3$-threshold extends to several generalized automatic classes, while suggesting the limits and computational aspects of such constructions; a general theorem for all generalized automatic families remains open.

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.

Figures (2)

• Figure 1: Fibonacci automaton computing the sequence $\bf y$.
• Figure 2: Tribonacci automaton computing a squarefree sequence.

Theorems & Definitions (26)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6