Algorithmic randomness and Fourier analysis

Suppose $1 < p < \infty$. Carleson's Theorem states that the Fourier series of any function in $L^p[-\pi, \pi]$ converges almost everywhere. We show that the Schnorr random points are precisely those that satisfy this theorem for every $f \in L^p[-\pi, \pi]$ given natural computability conditions on $f$ and $p$.