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

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.