Abstract
The Thue-Morse set $\mathcal{T}$ is the set of those non-negative integers whose binary expansions have an even number of $1$'s. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word
 $${\tt 0110100110010110\cdots},$$
 which is the fixed point starting with ${\tt 0}$ of the word morphism ${\tt 0\mapsto 01}$, ${\tt 1\mapsto 10}$. The numbers in $\mathcal{T}$ are commonly called the 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.
Highlights
A subset X of N is said to be b-recognizable if the base-b expansions of the elements of X form a regular language
The famous theorem of Cobham tells us that any non-trivial recognizable property of numbers is dependent on the base we choose: the only sets that are b-recognizable for all bases b are the finite unions of arithmetic progressions [19]
Since the finite unions of arithmetic progressions are precisely the subsets of N that are definable by first order formulas in the Presburger arithmetic N, +, this characterization provides us with a logical interpretation of Cobham’s theorem
Summary
A subset X of N is said to be b-recognizable if the base-b expansions of the elements of X form a regular language. Since the finite unions of arithmetic progressions are precisely the subsets of N that are definable by first order formulas in the Presburger arithmetic N, + , this characterization provides us with a logical interpretation of Cobham’s theorem. This result turned out to be a powerful tool for showing that many properties of b-automatic sequences are decidable and, further, that many enumeration problems of b-automatic sequences can be described by b-regular sequences in the sense of Allouche and Shallit [5, 7, 18]
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