Let $K\subset R^d$ be a compact set. It is well known that its Fourier dimension is defined by
$$\dim\_F K=\sup\{s\ge 0: \exists \mu \in M\_1(K) s.t. \hat{\mu}(x)=O(|x|^{-s/2})\}(|x|\to\infty),$$
where $M\_1(K)$ denotes the set of probability measures with support in $K.$


My question is: Does the condition $\dim\_FK=0$ imply that $K$ can not support a measure $\mu$ with decaying Fourier transformation? In other word, there is no Rajchman measure spported on $K?$ 


Maybe there exists measure with decaying Fourier transformation according other ``speed" rather than polynomial speed. Can somebody give an example? Many thanks.



Does zero Fourier dimension imply there is no Rajchman measure?
No, this does not follow. Since the Hausdorff dimension dominates the Fourier dimension, it suffices to establish the existence of a compact set $K$ of Hausdorff dimension zero that supports a Rajchman measure.


The last theorem of section 3 (attributed to Ivashev-Musatov 1962) of [Lyons's survey](https://pdfs.semanticscholar.org/9640/509efbee86d4d80b9f9a40685ffda2484fa1.pdf) gives a considerably stronger version of this statement (note that $\mathcal U\_0$ is defined in the introduction of the paper as the collection of sets that are annihilated by all Rajchman measures).


