Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.


share|cite|improve this question
What do you mean by "$K$ contains an interior point" because obviously $[0,1]$ does contain more than one interior point! – Mercy King 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 King 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.