  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
$m\mathcal{T}+r$.
