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.

翻译：我们研究一种源于整数3/2进制展开的二元Thue--Morse型序列，这是有理基数计数系统中典型的自动序列。由于该序列由态射的周期迭代而非单一本原替换生成，经典的Perron--Frobenius方法无法直接用于确定符号频率。我们证明符号${\tt 0},{\tt 1}$均以频率$1/2$出现，并展示了其因子集的均匀递归性与对称性。该证明揭示了词组合学与调和分析之间的结构桥梁：一阶差分序列被证明具有Toeplitz性质，从而提供动力学刚性，而滤波频率则自然地编码了可提升至2进整数紧群的二进结构。在此2进设定下，解替换成为傅里叶系数上的线性算子，谱压缩论证确保了极限密度的唯一性。我们的结果解答了Dekking关于兄弟序列的若干猜想，并展示了紧群上的调和分析如何能与替换动力学有效结合。