Let q be a natural number \(\geq 2\) and let \(e_ k(n)\in \{0,1,...,q-1\}\) be the k-th digit of the q-adic representation of n. Let \(G_ q\) be the set of nonnegative integers endowed with the group law \(\oplus\) defined as the addition to base q without carry, i.e., \(e_ k(m\oplus n)=e_ k(m)+e_ k(n)\) (mod 1). For any set \({\mathcal A}\) the group \(G_ q\) acts like a shift T on the set \({\mathcal A}^ N\) of \({\mathcal A}\)-valued sequences, namely: \(T_ g(u)_ n:=u_{n\oplus g}\), for n,g\(\geq 0.\)
In the paper under review, the authors study the set \({\mathcal M}_ q({\mathcal A})\) of periodic points of the action T. It is worth to notice (Lemma 2) that for every \(u\in {\mathcal M}_ q({\mathcal A})\) the image u(\({\mathbb{N}})\) is finite. Moreover, there exists a map \(\psi: {\mathbb{C}}\to {\mathcal A}\) and \(v\in {\mathcal M}_ q({\mathbb{C}})\) such that \(u=\psi \circ v\). Basic examples (for \({\mathcal A}={\mathbb{C}})\) are characters of \(G_ q\). Hence the set \({\mathcal M}_ q({\mathbb{C}})\) is typical and in fact (Theorem 3) \({\mathcal M}_ q({\mathbb{C}})\) is just the complex vector space generated by all characters of \(G_ q\). Let E be a subset of \({\mathbb{N}}\) and define \(s_ E(n):=\sum_{k\in E}e_ k(n)\). Then \(\chi_{E,q}(n):=\exp (2\pi is_ E(n)/q)\) is a character of \(G_ q.\)
Another characterisation of \({\mathcal M}_ q({\mathcal A})\) is derived from the fact that any character \(\psi\) on \(G_ q\) has the form \(\psi:=\chi_ 1...\chi_ d\) \((\chi_ j:=\chi^{t_ j}_{E_ j,q})\) where \(E_ j\subset {\mathbb{N}}\) and \(t_ j\in {\mathbb{Z}}\), for \(1\leq j\leq d\). It is well known that \(\psi_{E,q}\) is a generalized Morse sequence in the sense of M. Keane [Z. Wahrsch. Verw. Geb. 10, 335-353 (1968; Zbl 0162.072)] and such a sequence has a correlation which is the Fourier transform of the so-called spectral measure, denoted here by \(\mu_{E,q}\). The authors prove that for \(q\neq q'\) and subsets E, \(E'\) of \({\mathbb{N}}\) the measures \(\mu_{E,q}\) and \(\mu_{E',q'}\) are mutually singular. In the case \(q=q'\), these measures are mutually singular if and only if the symmetric difference \(E\Delta E'\) is infinite. The most difficult case corresponds to \(q=q'=2\) and was previously studied by S. Kakutani [Proc. Fifth Berkeley Symp., Vol. 2, 405-414 (1967; Zbl 0217.380)], M. Keane [Invent. Math. 16, 309-324 (1972; Zbl 0241.28014)] and the reviewer [Acta Arith. 55, 119-135 (1990; Zbl 0716.11038)].