Log-integrability of Rademacher Fourier series, with applications to random analytic functions

TL;DR

This paper proves that powers of the logarithm of Rademacher Fourier series are integrable and applies this to study the distribution of values of random Taylor series, addressing a long-standing question.

Contribution

It introduces a new integrability result for logarithmic powers of Rademacher Fourier series and applies it to analyze the value distribution of random Taylor series.

Findings

Logarithmic powers of Rademacher Fourier series are integrable.

Application to the distribution of values of random Taylor series.

Answers a long-standing question by J.-P. Kahane.

Abstract

We prove that any power of the logarithm of Fourier series with random signs is integrable. This result has applications to the distribution of values of random Taylor series, one of which answers a long-standing question by J.-P. Kahane.