The Fourier transform of the Cantor set is just zero. (It has zero Lebesgue measure.) For a subset $E \subseteq S^1$, the measure $\mu = \chi_E dz$ is absolutely continuous, so the Fourier coefficients are Cesaro convergent converge to 0 (I think this known as Wiener's $0$. This is just the Riemann--Lebesgue Lemma.Google seems to disagree.)
More interesting is what happens with the Cantor measure $\mu$. Then $\limsup_{n \to \infty} |\hat{\mu}(n)| > 0$. (Easy!).
The main difference between these two cases is the mass of the measure ...
