  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 \[ \lim_{c\to +0} \varlimsup_{t\to \infty}\frac{1}{t}\log
(\mathrm{Leb}\{x\in [-e^t, e^t]\mid |\mathcal{F}\mu(\xi)| \ge e^{-ct} \})=0. \]

Dynamical Systems