The Thue-Morse sequence in base 3/2

TL;DR

This paper explores the properties of the Thue-Morse sequence in base 3/2, demonstrating its fixed point nature under certain substitutions and analyzing its invariance and conjectured properties.

Contribution

It introduces a novel analysis of the base 3/2 representation of natural numbers, showing the fixed point behavior of digit sum functions and their invariance properties.

Findings

Sum of digits function is a fixed point of a 2-block substitution.

The sequence modulo 2 is mirror invariant.

Comparison with a variant of base 3/2 representation and general results on p-q block substitutions.

Abstract

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 $x_{3/2}$ of a 2-block substitution on $\{0,1\}$. We prove that $x_{3/2}$ is mirror invariant, and present a list of conjectured properties of $x_{3/2}$, 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.