On a theorem of H. Daboussi (Q2714168)

For arbitrary arithmetic functions \(f\) and \(g\) let NEWLINE\[NEWLINEM(f,g)=\limsup_{x\to\infty}x^{-1}\biggl|\sum_{n\leq x}f(n)g(n)\biggr|.NEWLINE\]NEWLINE For real \(\alpha\), let \(e_\alpha(n)=\exp(2\pi\text{in}\alpha)\). The theorem referred to in the title shows that if \(f\) is a multiplicative function and \(|f(n)|\leq 1\) then \(M(f,e_\alpha)=0\) for each irrational \(\alpha\) [cf. \textit{H. Daboussi} and \textit{H. Delange}, C. R. Acad. Sci., Paris, Sér. A 278, 657-660 (1974; Zbl 0292.10034); J. Lond. Math. Soc. (2) 26, 245-264 (1982; Zbl 0499.10052)]. An arithmetical function \(f\) is said to be uniformly summable if \(\lim_{y\to \infty}(\sup_{x\geq 1}x^{-1}\sum_{n\leq x,f(n)|\geq y}|f(n)|)=0\). For a fixed integer \(q\geq 2\), an arithmetical function \(g\) is called \(q\)-multiplicative if \(g(0)=1\) and, for \(q\)-ary expansions, \(g(\sum_{j\geq 0}\varepsilon_j q^j)=\prod_{j\geq 0}g(\varepsilon_j g^j)\) provided \(\varepsilon_j\in \{0,1,\dots,q-1\}\).NEWLINENEWLINENEWLINEThe main result is the following generalization of Daboussi's theorem:NEWLINENEWLINENEWLINEIf \(f\) is a uniformly summable multiplicative function, \(g\) is a \(q\)-multiplicative function of modulus 1 and \(M(f,g)>0\) then there exist rational numbers \(\beta\) and \(\gamma\) such that \(g(n)=e_\beta(n)h(n)\) with \(\sum^\infty_{j=0}\sum^{q-1}_{c=0} \text{Re} (1-h(cq^j))<\infty\) and \(\gamma\) belongs to the Bohr--Fourier spectrum of \(f\), i.e., \(M(f,e_{-\gamma})>0\). As an application, uniform value distribution mod~1 is obtained for the sum of a real-valued additive function and certain \(q\)-additive function.