Fourier decay bound and differential images of self-similar measures

Abstract

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 $\rho \in (0,1/m)$, 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 Erdős, which is important in estimating the oscillatory integrals. An application to the existence of normal numbers in fractals is also given.

9 pages

Cited by in corpus (1)

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