On the Fourier Transforms of Nonlinear Self-similar Measures

In-homogeneous self-similar measures can be viewed as special cases of nonlinear self-similar measures. In this paper, we study the asymptotic behaviour of the Fourier transforms of nonlinear self-similar measures. Some typical examples are exhibited, and we show that the Fourier transforms of those measures are usually localized, i.e., the Fourier transforms decay rapidly at ∞ . We also discuss the infinity lower Fourier dimension of in-homogeneous self-similar measures and obtain its non-trivial bounds. The result confirms Conjecture 2.3 in Olsen and Snigireva (Math Proc Camb Philos Soc 144:465–493, 2008).