Minimal automaton for multiplying and translating the Thue-Morse set

Abstract

The Thue-Morse set $\mathcal{T}$ is the set of those non-negative integers whose binary expansions have an even number of $1$. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word ${\tt abbabaabbaababba\cdots}$, which is the fixed point starting with ${\tt a}$ of the word morphism ${\tt a\mapsto ab,b\mapsto ba}$. The numbers in $\mathcal{T}$ are commonly called the {\em evil numbers}. We obtain an exact formula for the state complexity of the set $m\mathcal{T}+r$ (i.e.\ the number of states of its minimal automaton) with respect to any base $b$ which is a power of $2$. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all $2^p$-expansions of the set of integers $m\mathcal{T}+r$ for any positive integers $p$ and $m$ and any remainder $r\in\{0,\ldots,m-1\}$. The proposed method is general for any $b$-recognizable set of integers. As an application, we obtain a decision procedure running in quadratic time for the problem of deciding whether a given $2^p$-recognizable set is equal to a set of the form $m\mathcal{T}+r$.