Algorithmic randomness and Fourier analysis

TL;DR

This paper characterizes Schnorr random points as exactly those points where Fourier series convergence, as stated in Carleson's Theorem, holds for all computably well-behaved functions in L^p spaces.

Contribution

It establishes a novel connection between algorithmic randomness and Fourier analysis, identifying Schnorr randomness as the key condition for convergence.

Findings

Schnorr random points satisfy Fourier series convergence for all computable L^p functions.

The result links algorithmic randomness with classical harmonic analysis.

Provides a new perspective on randomness through Fourier analysis in L^p spaces.

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 $p$.