Piatetski-Shapiro phenomenon in the uniqueness problem

We extend the phenomenon discovered by Piatetski-Shapiro (1954) to $l^q$ spaces. To be precise, for any $q>2$ we construct a compact K on the circle, which supports a distribution S with Fourier transform $\stackrel{?}{S}\in l^q$, but does not support such a measure. To cite this article: N. Lev, A. Olevskii, C. R. Acad. Sci. Paris, Ser. I 340 (2005).