The Thue–Morse sequence in Base 3/2

We discuss the base 3/2 representation of the natural numbers. We prove that the sum-of-digits function of the representation is a fixed point of a 2-block substitution on an infinite alphabet, and that this implies that sum-of-digits function modulo 2 of the representation is a fixed point x3/2 of a 2-block substitution on {0,1}. We prove that x3/2 is invariant for taking the binary complement, and present a list of conjectured properties of x3/2, which we think will be hard to prove. Finally, we make a comparison with a variant of the base 3/2 representation, and give a general result on p-q-block substitutions.