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

Given any function $\phi \text{\hspace{0.17em}⁣}:[0,\mathrm{\infty })\to (0,1]$ satisfying ${\lim }_{\xi \to \mathrm{\infty }}\phi (\xi )=0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\mathrm{\infty }}$ self-conformal measures which are Rajchman and whose Fourier transform $\widehat{\mu }$ satisfies $$\underset{\xi \to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\mathrm{\mid }\widehat{\mu }(\xi )\mathrm{\mid }}{\phi (\xi )}>0.$$ 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 $({10}^{n}x\mathrm{m}\mathrm{o}\mathrm{d}1{)}_{n=1}^{\mathrm{\infty }}$ equidistributes extremely slowly.