State Complexity of Shifts of the Fibonacci Word

TL;DR

This paper investigates the state complexity of automata generating shifted Fibonacci sequences, showing it is logarithmic in the shift amount, which approaches the theoretical minimum for such aperiodic sequences.

Contribution

It introduces bounds on the state complexity of shifted Fibonacci word automata, combining combinatorial and Diophantine approximation techniques.

Findings

State complexity of shifted sequences is O(log c)

Automata for shifted sequences are near minimal in size

Techniques involve a mix of automata theory and number theory

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.