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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!

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What do you mean by "$K$ contains an interior point" because obviously $[0,1]$ does contain more than one interior point! –  Mercy Jun 22 '12 at 14:31
    
@Mercy Your statement isn't true, the Cantor set is a counterexample. There exist even variants of the Cantor set with positive Lebesgue measure, take the Smith–Volterra–Cantor set for example. –  Thomas Klimpel Jun 23 '12 at 21:32
    
@ThomasKlimpel You're right! Thanks –  Mercy Jun 23 '12 at 21:40
    
Have you tried the case when $K$ is a fat Cantor? –  Davide Giraudo Aug 8 '12 at 11:14
    
@DavideGiraudo: I haven't tried. Do you have any idea? –  Syang Chen Aug 8 '12 at 16:31

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