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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!2012-06-22
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    @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.2012-06-23
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    @ThomasKlimpel You're right! Thanks2012-06-23
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    Have you tried the case when $K$ is a fat Cantor?2012-08-08
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    @DavideGiraudo: I haven't tried. Do you have any idea?2012-08-08
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    Maybe with Lebesgue decomposition measure, we have to look at measure written as a sum of an absolutely continuous measure (with respect to Lebesgue measure $\lambda$) and a singular measure (wrt $\lambda$), the discrete part won't help. I will try to do the computation in the case of a fat Cantor.2012-08-08

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