Symbols frequencies in the Thue--Morse word in base $3/2$ and related conjectures

TL;DR

This paper investigates the symbol frequencies and structural properties of a Thue--Morse-type sequence generated from base-$3/2$ expansion, revealing uniform distribution, recurrence, and symmetry through harmonic analysis and dynamical systems techniques.

Contribution

It introduces a novel analysis of a rational-base automatic sequence using harmonic analysis on compact groups, establishing symbol frequencies and structural properties that were previously conjectural.

Findings

Both symbols occur with frequency 1/2.

The sequence exhibits uniform recurrence and symmetry.

Harmonic analysis provides a new approach to substitution dynamics.

Abstract

We study a binary Thue--Morse-type sequence arising from the base-$3/2$ expansion of integers, an archetypal automatic sequence in a rational base numeration system. Because the sequence is generated by a periodic iteration of morphisms rather than a single primitive substitution, classical Perron--Frobenius methods do not directly apply to determine symbol frequencies. We prove that both symbols ${\tt 0},{\tt 1}$ occur with frequency $1/2$ and we show uniform recurrence and symmetry properties of its set of factors. The proof reveals a structural bridge between combinatorics on words and harmonic analysis: the first difference sequence is shown to be Toeplitz, providing dynamical rigidity, while filtered frequencies naturally encode a dyadic structure that lifts to the compact group of $2$-adic integers. In this $2$-adic setting, desubstitution becomes a linear operator on Fourier coefficients, and a spectral contraction argument enforces uniqueness of limiting densities. Our results answer several conjectures of Dekking (on a sibling sequence) and illustrate how harmonic analysis on compact groups can be fruitfully combined with substitution dynamics.