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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, andWang (obtained simultaneously by Baker and Banaji), showing that anyself-conformal measure with respect to a non-affine real analytic IFS haspolynomial Fourier decay.

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