  R. Kaufman and M. Tsujii proved that the Fourier transform of self-similar
measures has a power decay outside of a sparse set of frequencies. We present a
version of this result for homogeneous self-similar measures, with quantitative
estimates, and derive several applications: (1) non-linear smooth images of
homogeneous self-similar measures have a power Fourier decay, (2) convolving
with a homogeneous self-similar measure increases correlation dimension by a
quantitative amount, (3) the dimension and Frostman exponent of (biased)
Bernoulli convolutions tend to $1$ as the contraction ratio tends to $1$, at an
explicit quantitative rate.
