Minimal Automaton For Multiplying And Translating The Thue-Morse Set

The Thue-Morse set 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 word0110100110010110...,which is the fixed point starting with 0 of the word morphism 0 bar right arrow 01, 1 bar right arrow 10. The numbers in T are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set mT + 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 mT + r for any positive integers p and m and any remainder r is an element of {0, ..., m-1}. The proposed method is general for any b-recognizable set of integers.