Abstract: A point in a probability space is called algorithmically random if it has no rare measure-theoretic properties that are simple to define via computability theory; the set of all such points will have measure 1 for any reasonable definition of randomness. Carleson's Theorem states that the Fourier series of certain integrable functions converge on a set of measure 1 as well. I will consider this theorem in the context of computable analysis and show that the points that satisfy this version of Carleson's Theorem are precisely the Schnorr random points.