  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 $( x^n
\mod 1 )_{n\geq 1}$ 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.
