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

Abstract

Given any function $\ddot{I}•:[0,\infty )\to (0,1]$ satisfying ${\lim }_{\hat{I}¾\to \infty }\ddot{I}•(\hat{I}¾)=0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty }$ self-conformal measures which are Rajchman and whose Fourier transform $\hat{\hat{I}}¼$ satisfies \[ \limsup_{ξ\to\infty}\frac{|\widehatμ(ξ)|}{ϕ(ξ)}>0.\] Moreover, we derive new sufficient conditions for a self-conformal measure to be Rajchman, and construct an explicit self-similar measure $\hat{I}¼$ such that $\hat{I}¼$ almost every $x$ is normal in base $10$ but the sequence $(10^{n}x\mathrm{mod}1{)}_{n=1}^{\infty }$ equidistributes extremely slowly.