Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the Fourier-Stieltjes transform of $\mu$, has decay $O(|\xi|^{-1})$?
Note that when $K=[0,1]$, we can simply take $\mu=\chi_{[0,1]}dt$, see this post. Generally, if $K$ contains an interior point, then by the same token such a probability measure trivially exists. But things become unclear to me when $K$ is a general set.
Thanks!