The Lower Fourier Dimensions of In-Homogeneous Self-similar Measures

Abstract

The in-homogeneous self-similar measure $\mu$ is defined by the relation

$$\begin{aligned} \mu =\sum _{j=1}^N p_j \mu \circ S_j^{-1}+p\nu , \end{aligned}$$

where $(p_1,\ldots ,p_N,p)$ is a probability vector, each $S_j:\mathbb {R}^d\rightarrow \mathbb {R}^d$, $j=1,\ldots ,N$, is a contraction similarity, and $\nu$ is a Borel probability measure on $\mathbb {R}^d$ with compact support. In this paper, we study the asymptotic behavior of the Fourier transforms of in-homogeneous self-similar measures. We obtain non-trivial lower and upper bounds for the qth lower Fourier dimensions of the in-homogeneous self-similar measures without any separation conditions. Moreover, if the IFS satisfies some separation conditions, the lower bounds for the qth lower Fourier dimensions can be improved. These results confirm conjecture 2.5 and give a positive answer to the question 2.7 in Olsen and Snigireva’s paper (Math Proc Camb Philos Soc 144(2):465–493, 2008).