Algorithmic randomness and Fourier analysis

Abstract

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 $p$.