State complexity of the multiples of the Thue-Morse set

TL;DR

This paper derives an exact formula for the state complexity of multiplying the Thue-Morse set by a constant in bases that are powers of two, providing explicit minimal automata constructions.

Contribution

It introduces a precise formula for state complexity of multiples of the Thue-Morse set in power-of-two bases and constructs minimal automata explicitly.

Findings

Exact state complexity formula for multiples of Thue-Morse set

Explicit minimal automata for all multiples in power-of-two bases

Constructive proof enabling automaton construction

Abstract

The Thue-Morse set is the set of those nonnegative integers whose binary expansions have an even number of $1$. We obtain an exact formula for the state complexity of the multiplication by a constant of the Thue-Morse set $\mathcal{T}$ with respect with 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 $m\mathcal{T}$ for any positive integers $m$ and $p$.