Self-similar and self-conformal measures with slow Fourier decay

Abstract

Given any function $\phi \colon [0,\infty)\to (0,1]$ satisfying $\lim_{\xi\to\infty}\phi(\xi) = 0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-conformal measures which are Rajchman and whose Fourier transform $\widehat{\mu}$ satisfies \[ \limsup_{\xi\to\infty}\frac{|\widehat{\mu}(\xi)|}{\phi(\xi)}>0.\] Moreover, we derive new sufficient conditions for a self-conformal measure to be Rajchman, and construct an explicit self-similar measure $\mu$ such that $\mu$ almost every $x$ is normal in base $10$ but the sequence $(10^{n}x \mod 1)_{n=1}^{\infty}$ equidistributes extremely slowly.