The Thue-Morse sequence in base 3/2

zbMath1510.11086arXiv2301.13563MaRDI QIDQ6424944

F. Michel Dekking

Publication date: 31 January 2023

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.

Mathematics Subject Classification ID

Radix representation; digital problems (11A63) Automata sequences (11B85)

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