Ergodicité et pureté des produits de Riesz. (Ergodicity and purity of Riesz products) (Q1263031)

It is shown that Riesz products on the torus are ergodic measures under a lacunarity condition for their frequencies, without any arithmetic requirement, and that the given condition is the best possible from this viewpoint. A similar criterion is given for the purity property previously discussed by \textit{B. Host} and the author [C. R. Acad. Sci., Paris, Sér. I 306, 409-412 (1988; Zbl 0653.22006)] and this provides an example of a pure measure which is singular to all its translates and henceforth not ergodic.