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 $\mu$. As an application, we show that the correlations of all finite orders of $(x^n\mathrm{mod}1{)}_{n\ge 1}$ 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.