# Fourier coeffients of Cantor measure

For $$0<\theta<\frac{1}{2}$$, denote by $$\mu_\theta$$ the uniform Cantor measure with dissection ratio $$\theta$$. It is not hard to show that the Fourier–Stieltjes transform of $$\mu_\theta$$ is $$\widehat{\mu}_\theta(\xi)=\prod_{k=1}^{\infty} \cos(\pi\theta^k\xi)$$ (up to scaling and constant multiple). It is well known that $$\widehat{\mu}_\theta(\xi)\to 0$$ as $$\xi\to\infty$$ if and only if $$\theta$$ is a Pisot number.

Now we are concentrated on the case where $$1/\theta$$ is an integer. We know that $$\limsup_{n\in \mathbb{Z}, n\to\infty} \widehat{\mu}_\theta(n)>0$$. It is not hard to see that $$\liminf_{n\in \mathbb{Z}, n\to\infty} \widehat{\mu}_\theta(n)=0$$. My question is whether we can characterize (or approximate) the set $$A_{\epsilon}:=\{n\in \mathbb{Z}: |\widehat{\mu}_\theta(n)|>\epsilon\},$$ for $$0<\epsilon<1$$. My second question is what the "maximal" set $$B$$ is such that $$\liminf_{n\in \mathbb{Z}\setminus B, n\to\infty} \widehat{\mu}_\theta(n)>0.$$ Thanks.