On the Fourier transforms of self-similar measures

For the Fourier transform $\mathcal{F}\mu$ of a general (non-trivial) self-similar measure $\mu$ on the real line $\mathbb{R}$, we prove a large deviation estimate $$\underset{c\to +0}{\lim }\underset{t\to \mathrm{\infty }}{\mathrm{}}\frac{1}{t}\log (\mathrm{L}\mathrm{e}\mathrm{b}\{x\in [-e^t,e^t]\mid \mathrm{\mid }\mathcal{F}\mu (\xi )\mathrm{\mid }\ge e^{-ct}\})=0.$$