The Thue–Morse Sequence in Base 3/2

Journal article (2023)

Authors

F.M. Dekking Applied Probability - EEMCS , Centrum Wiskunde & Informatica (CWI)
Research Group
Applied Probability (EEMCS) (TU Delft)
Copyright: © 2023 F.M. Dekking
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Published Date

2023

Language

English

Reuse Rights

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

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 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.

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