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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.

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