Let \(d(n)\) be the sum of binary digits of \(n=0,1,\dots\) and let \(t_ n=(-1)^{d(n)}\) be the \(n\)-th term of the well-known Thue-Morse sequence. The asymptotic behavior of sums \(S_ i(n)= \sum_{\textstyle{{{0\leq k<n} \atop {k\equiv i\pmod p}}}} t_ k\), \(p\) odd prime, is studied. Let \(S(n)\) be the column vector with entries \(S_ i(n)\). The paper is based on the relation \(S(2^ s n)=MS(n)\) where \(s\) is the multiplicative order of \(2\bmod p\) and \(M\) is a suitable matrix. The authors show that there exists a continuous function \(F: [0,+\infty)\to \mathbb R^ p\) satisfying the scaling property \(F(2^ s x)=MF(x)\) such that \(S(n)=F(n)+O(1)\).
As a consequence (Theorem 5.1) \(S(n)=n^ \alpha \Phi(\log n/rs\log 2)+ \varepsilon(n)\) where \(\Phi: \mathbb R\to\mathbb R^ p\) is continuous of period 1 and \(\alpha= \log\lambda_ 1/s \log 2\) where \(\lambda_ 1\) is the largest magnitude of eigenvalues of \(M\). Moreover \(\varepsilon(n)\in O(n^ \beta)\) with \(\beta<\alpha\).
Previous works have been done by several authors, in particular by J. Coquet for \(p=3\) [Invent. Math. 73, 107–115 (1983; Zbl 0528.10006)] and the general case (where \(p\) is any integer \(\geq 3\)) has been investigated by J.-M. Dumont [Discrépance des progressions arithmétiques dans la suite de Morse, Thèse 3ième cycle, Univ. Aix- Marseille 1 (1984), see also C. R. Acad. Sci., Paris, Sér. I 297, 145–148 (1983; Zbl 0533.10005)]. The authors also extend their results to any base \(b>2\).