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

We prove a van der Corput lemma for non-atomic self-similar measures $μ$. 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 $μ$-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.

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