2 added 18 characters in body

I am wondering how badly summable the Fourier transform of the characteristic function of a measurable subset of $S^1$ can be.

Question: Let $\alpha \colon \mathbb N \to [1,\infty)$ be a monotone increasing function with $\lim_{n \to \infty} \alpha(n) = \infty$. Is there a measurable subset $E \subset S^1$, such that $$\sum_{n \in \mathbb Z} | \widehat \chi_E(n)|^2 \cdot \alpha(|n|) = \infty \ ?$$ Here, $\widehat \chi(n)$ are the usual moments $$\widehat \chi_E(n):= \int_E z^n \ dz.$$

The only example I know is the Fourier transform of the characteristic function of an interval, which grows like $1/n$. On the other hand, one can easily see that the growth cannot be better than $1/n$ (something like $1/n^{1 + \varepsilon}$), since $\ell^1 \mathbb Z \subset C(S^1)$.

More concretely:

Question: Can anyone compute the growth of the Fourier transform of the characteristic function of something like the a Cantor set of non-zero measure?

Again, more abstractly:

Question: What can be said about the growth of the Fourier transform of the characteristic function of a generic subset of $S^1$?

1

# Fourier transforms of characteristic functions

I am wondering how badly summable the Fourier transform of the characteristic function of a measurable subset of $S^1$ can be.

Question: Let $\alpha \colon \mathbb N \to [1,\infty)$ be a monotone increasing function with $\lim_{n \to \infty} \alpha(n) = \infty$. Is there a measurable subset $E \subset S^1$, such that $$\sum_{n \in \mathbb Z} | \widehat \chi_E(n)|^2 \cdot \alpha(|n|) = \infty \ ?$$ Here, $\widehat \chi(n)$ are the usual moments $$\widehat \chi_E(n):= \int_E z^n \ dz.$$

The only example I know is the Fourier transform of the characteristic function of an interval, which grows like $1/n$. On the other hand, one can easily see that the growth cannot be better than $1/n$ (something like $1/n^{1 + \varepsilon}$), since $\ell^1 \mathbb Z \subset C(S^1)$.

More concretely:

Question: Can anyone compute the growth of the Fourier transform of the characteristic function of something like the Cantor set?

Again, more abstractly:

Question: What can be said about the growth of the Fourier transform of the characteristic function of a generic subset of $S^1$?