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

The in-homogeneous self-similar measure μ is defined by the relation μ = ∑ j = 1 N p j μ ∘ S j - 1 + p ν , where ( p 1 , … , p N , p ) is a probability vector, each S j : R d → R d , j = 1 , … , N , is a contraction similarity, and ν is a Borel probability measure on 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 q th 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 q th 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).