On universality and convergence of the Fourier series of functions in the disc algebra

TL;DR

This paper constructs functions in the disc algebra with Fourier series that are pointwise universal on dense G-delta sets and have divergence sets of Hausdorff dimension zero, exploring universality and divergence properties.

Contribution

It introduces functions with universal Fourier series on complex sets and analyzes divergence behavior on measure-zero sets, advancing understanding of Fourier series in the disc algebra.

Findings

Existence of functions with pointwise universal Fourier series on dense G-delta sets.

Divergence sets of these Fourier series can have Hausdorff dimension zero.

Certain measure-zero sets do not support uniformly universal Fourier series.

Abstract

We construct functions in the disc algebra with pointwise universal Fourier series on sets which are G-delta and dense and at the same time with Fourier series whose set of divergence is of Hausdorff dimension zero. We also see that some classes of closed sets of measure zero do not accept uniformly universal Fourier series, although all such sets accept divergent Fourier series.