  We survey some of our recent results on the geometry of spatially independent
martingales, in a more concrete setting that allows for shorter, direct proofs,
yet is general enough for several applications and contains the well-known
fractal percolation measure. We study self-convolutions and Fourier decay of
measures in our class, and present applications of these results to the
restriction problem for fractal measures, and the connection between arithmetic
structure and Fourier decay.
