On the Fourier transforms of self-similar measures

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

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