TITLE: Fourier decay rate of Cantor measures
QUESTION [14 upvotes]: For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known that $C_\theta$ carries a uniform measure $\mu_\theta$ which is usually called Cantor measure. And it is not hard to show that the Fourier–Stieltjes transform of $\mu_\theta$ is (up to scaling and constant multiple)
$$\hat{\mu}_\theta(\xi)=\prod^{\infty}_{k=1} \cos(\theta^k\xi)$$ 
But unlike integrable function we do not have Riemann-Lebesgue lemma for these Cantor measures. In fact a theorem of Erdős and Salem says $\hat{\mu}_\theta (\xi)=o(1)$ as $|\xi|\rightarrow\infty$ if and only if $\theta^{-1}$ is a not PV number. On the other hand, it is known that for some $\theta^{-1}$ not a PV number, $\hat{\mu}_\theta (\xi)$ does not decay at any positive rate, even though $\hat{\mu}_\theta (\xi)=o(1)$.
My question is whether there exists $\theta$ such that $\hat{\mu}_\theta (\xi)=O(|\xi|^{-\alpha})$ for some $\alpha>0$? How much is known about the precise decay rate of $\hat{\mu}_\theta (\xi)$? 
Thanks in advance.
Edit: To make the second question precise, I was actually wondering if there exists Salem set (as pointed out by Pablo) among these Cantor sets. So the rate of decay I was expecting is (or arbitrarily close to) half of the Hausdorff dimension $\frac{1}{2}\dim_H(C_\theta)=\frac{1}{2}\frac{\log(1/2)}{\log(\theta)}$.

REPLY [3 votes]: I would start looking here:
MR1785620 (2001m:42020)
Peres, Yuval; Schlag, Wilhelm; Solomyak, Boris,
Sixty years of Bernoulli convolutions.
Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), 39–65, 
Progr. Probab., 46, Birkhäuser, Basel, 2000.