State Complexity of Shifts of the Fibonacci Word

Abstract

The Fibonacci infinite word ${\bf f} = (f_i)_{i \geq 0} = 01001010\cdots$ is one of the most celebrated objects in combinatorics on words. There is a simple $5$-state automaton that, given $i$ in lsd-first Zeckendorf representation, computes its $i$'th term $f_i$, and a $2$-state automaton for msd-first. In this paper we consider the state complexity of the automaton generating the shifted sequence $(f_{i+c})_{i \geq 0}$, and show that it is $O(\log c)$ for both msd-first and lsd-first input. This is close to the information-theoretic minimum for an aperiodic sequence. The techniques involve a mixture of state complexity techniques and Diophantine approximation.

Figures (5)

• Figure 1: DFAOs for the Fibonacci word $\bf f$. Msd-first on the left; lsd-first on the right.
• Figure 2: Illustrations of the partition lemma and construction of the automaton for $c=10$.
• Figure 3: Lsd-first DFAO for $(f(i+10))_{i \geq 0}$.
• Figure 4: Msd-first DFAO for $(f(i+10))_{i \geq 0}$.
• Figure 5: Lsd-first DFAO for $(f(i+12))_{i \geq 0}$.

Theorems & Definitions (28)

• lemma 1
• proof
• lemma 2
• proof
• theorem 1
• definition 1
• lemma 3
• remark 1
• remark 2
• proof : of \ref{['theorem:sc-lsd']}