On a sequence of Kimberling and its relationship to the Tribonacci word

TL;DR

The paper analyzes Kimberling's sequence ${\bf B}$, defined as the infinite fixed point of the inflation rules $00 \rightarrow 0101$, $1 \rightarrow 10$, and proves Kimberling's conjectures about its length sequence and related sequences. It uncovers a tight connection to the Tribonacci word ${\bf TR}$ via the morphisms ${\pi}$ and ${\varphi}$, enabling an automata-theoretic treatment and Walnut-based proofs for key properties. The authors establish that ${\bf B}$ has subword complexity ${\rm comp}(n)=2n$ for $n\ge 1$ and determine its critical exponent ${\rm ce}({\bf B}) = 2 + \frac{1}{\psi-1}$ with $\psi$ the real root of $X^3 - X^2 - X - 1$, approximately $3.19148788395$. They also analyze bispecial factors and return words to provide a rigorous framework for these results and demonstrate a methodology applicable to similar inflation sequences.

Abstract

In 2017, Clark Kimberling defined an interesting sequence ${\bf B} = 0100101100 \cdots$ of $0$'s and $1$'s by certain inflation rules, and he made a number of conjectures about this sequence and some related ones. In this note we prove his conjectures using, in part, the Walnut theorem-prover. We show how his word is related to the infinite Tribonacci word, and we determine both the subword complexity and critical exponent of $\bf B$.

Figures (1)

• Figure 1: The Tribonacci automaton for ${\bf B}[n]$.

Theorems & Definitions (21)

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