Fourier decay bound and differential images of self-similar measures

In this note, we investigate $C^2$ differential images of the homogeneous self-similar measure associated with an IFS $$\mathcal{I}$={\rho x+a_j}_{j=1}^m$ satisfying the strong separation condition and a positive probability vector $\vec{p}$. It is shown that the Fourier transforms of such image measures have power decay for any contractive ratio $\rho \in (0,1\mathrm{/}m)$, any translation vector $\vec{a}=(a_1,\dots ,a_m)$ and probability vector $\vec{p}$, which extends a result of Kaufman on Bernoulli convolutions. Our proof relies on a key combinatorial lemma originated from Erd\H{o}s, which is important in estimating the oscillatory integrals. An application to the existence of normal numbers in fractals is also given.