Pointwise normality and Fourier decay for self-conformal measures

Abstract

Let $\Phi$ be a $C^{1+\gamma }$ smooth IFS on $\mathbb{R}$, where $\gamma >0$. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points $x$ that are absolutely normal. That is, for integer $p\ge 2$ the sequence $\{p^{k}x{\}}_{k\in \mathbb{N}}$ equidistributes modulo $1$. We thus extend several state of the art results of Hochman and Shmerkin about the prevalence of normal numbers in fractals. When $\Phi$ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless $\Phi$ has an explicit structure that is associated with some integer $n\ge 2$. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to $0$ at infinity. When $\Phi$ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.

V3: Revised according to comments made by the referee and Péter Varjú. To appear in Adv. Math