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.

翻译：斐波那契无限词${\bf f} = (f_i)_{i \geq 0} = 01001010\cdots$是词组合学中最著名的研究对象之一。存在一个简单的五状态自动机，给定以最低有效位优先的齐肯多夫表示输入$i$，计算其第$i$项$f_i$；而对于最高有效位优先输入，则存在一个两状态自动机。本文考虑生成移位序列$(f_{i+c})_{i \geq 0}$的自动机的状态复杂度，并证明对于最高有效位优先和最低有效位优先输入，该复杂度均为$O(\log c)$。这接近非周期序列的信息论下界。所采用的技术融合了状态复杂度方法与丢番图逼近。