This is an announcement for the paper "Thin sets of integers in Harmonic analysis and p-stable random Fourier series" by Pascal Lefevre, Daniel Li, Herve Queffelec, and Luis Rodriguez-Piazza. Abstract: We investigate the behavior of some thin sets of integers defined through random trigonometric polynomial when one replaces Gaussian or Rademacher variables by p-stable ones, with 1 < p < 2. We show that in one case this behavior is essentially the same as in the Gaussian case, whereas in another case, this behavior is entirely different. Archive classification: math.FA Mathematics Subject Classification: Primary: 43A46 ; secondary: 42A55; 42A61; 60G52 The source file(s), p-stableBETISfinale.tex: 72111 bytes, is(are) stored in gzipped form as 0902.2625.gz with size 21kb. The corresponding postcript file has gzipped size 143kb. Submitted from: lefevre@euler.univ-artois.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0902.2625 or http://arXiv.org/abs/0902.2625 or by email in unzipped form by transmitting an empty message with subject line uget 0902.2625 or in gzipped form by using subject line get 0902.2625 to: math@arXiv.org.