Van der Corput and metric theorems for geometric progressions for self-similar measures

Abstract We prove a van der Corput lemma for non-atomic self-similar measures. As an application, we show that the correlations of all finite orders of (xⁿ 1) ₍ ₁ converge to the Poissonian model for -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.