We prove a van der Corput lemma for non-atomic self-similar measures \(\mu \) . As an application, we show that the correlations of all finite orders of \(( x^n \mod 1 )_{n\ge 1}\) converge to the Poissonian model for \(\mu \) -a.e. x, assuming \(x>1\) . We also complete a recent result of Algom, Rodriguez Hertz, and Wang (obtained simultaneously by Baker and Banaji), showing that any self-conformal measure with respect to a non-affine real analytic IFS has polynomial Fourier decay.
