The author uses a theorem of \textit{J. Coquet} in Sur la mesure spectrale des suites \(q\)-multiplicatives [Ann. Inst. Fourier 29, 163-170 (1979; Zbl 0413.10046)], characterizing \(q\)-multiplicative functions of modulus 1 to be pseudorandom iff its Fourier-Bohr spectrum is empty, in order to show that the existence of such spectrum for the image by a character of an \(\mathbb{R}/\mathbb{Z}\)-valued \(q\)-additive function is equivalent to the vague convergence of a certain sequence of measures to a probability measure using also an argument of \textit{I. Ruzsa} [Lect. Notes Math. 928, 337-353 (1982; Zbl 0489.60012)], which works also for compact abelian groups. The results are extended to \(q\)-additive functions with values in locally compact abelian groups.