Fourier decay for self-similar measures

Abstract: We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity. In the homogeneous case, when all contraction ratios are equal, this is essentially due to Erd\H{o}s and Kahane. In the non-homogeneous case the difficulty we have to overcome is the apparent lack of convolution structure.