On the Fourier transforms of self-similar measures

Abstract

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

16 pages, 1 figure Several errors in the previous version are corrected. A few notations are changed for clearer exposition