rational base numeration systems; block substitutions; combinatorics on words; Thue–Morse sequence; frequency; uniform recurrence; Pontryagin duality; Fourier coefficients
Abstract :
[en] 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.
