A null series with small anti-analytic part. (Q1408113)

The authors present three significant theorems. Two of them are equivalent, the different ones read as follows: I. There is a power series \(F(z)= \sum c(n)z^n\) converging a.e. on the circle \(| z|= 1\) to some function \(f\in L^2(\mathbb{T})\), and \(f\) does not belong to the Hardy space \(H^2\). II. If a series \(\sum_{n\geq 0} c(n) e^{int}\) converges to \(f\in L^1(\mathbb{T})\) everywhere on \(\mathbb{T}\) outside some set of dimension \(< 1\) then it is the Fourier series of \(f\).