Existence of $L^\infty$ function on $\mathbb{T}$ whose Fourier series is $\ell^2$ but no better?

Question

I'm sure that this is classical--but can anyone provide a reasonable example of an $L^\infty(\mathbb{T})$ function whose Fourier series is $\ell^2$ but no better? Not even $L^2\log L$? Presumably one exists but nothing has come to my mind. I'm trying to understand just how far one can stretch a (version of a) particular conjecture on Boolean functions, the Fourier Entropy-Influence conjecture.

Answer 1

A theorem due to Kahane, Katznelson and de Leeuw says that for any sequence $(a_n)_{n\in \mathbb{Z}}$ belonging to $\ell_2$ there is a continuous function $f \in C(\mathbb{T})$ such that $|\widehat{f}(n)| \geqslant |a_n|$, so in general you don't get anything better than $\ell_2$. The proof can be found in Appendix B of Katznelson's book -- the construction is somewhat explicit.