Abstract :
Letqbe a positive integer,qgreater-or-equal, slanted2. In 1979, Coquet proved that aq-multiplicative function of modulus 1 is pseudo-random if and only if its Fourier–Bohr spectrum is empty. In this article, we show that a consequence of this result is that the existence of a Fourier–Bohr spectrum for the image by a character of a image/image-valuedq-additive functionfis equivalent to the existence of some constantαand a sequenceAnin image/image such that the sequence of measures 1/[x] ∑nless-than-or-equals, slantx δ(f(n)−αn−ALog q(x))converges vaguely to a probability measureνon image/image; we extend this result toq-additive functions with values in a locally compact abelian group.
