Carleson theorem (later extended by Hunt) states that given an $L^2$ function $f:{\mathbb R}/{\mathbb Z}\to{\mathbb C}$, the set of points $x$ where the Fourier series $$\lim_{n\to\infty}\sum_{k=-n}^n\hat f(k)e^{2\pi ik x}$$ does not converge to $f(x)$ has measure 0.

Kahane and Katznelson proved that given any measure zero set $E$ there is a continuous function $f:{\mathbb R}/{\mathbb Z}\to{\mathbb C}$ whose Fourier series diverges at all points of $E$.

These two results leave a little gap.

What is known about those sets $E$ for which there is an $L^2$ (or even continuous?) function $f$ whose Fourier series diverges at all points of $E$ and pointwise converges to $f$ at all points not in $E$? Can any measure zero set be such an $E$?