Singular distributions and symmetry of the spectrum

TL;DR

This survey explores the Fourier symmetry of measures and distributions on the circle, focusing on how their support size influences spectral symmetry, based on recent research and a symposium presentation.

Contribution

It consolidates recent findings on Fourier symmetry and support size relations, providing a comprehensive overview of the topic.

Findings

Fourier symmetry relates to support size of measures on the circle.

Distributions with certain support properties exhibit spectral symmetry.

The survey synthesizes recent advances and open questions in the field.

Abstract

This is a survey of the "Fourier symmetry" of measures and distributions on the circle in relation with the size of their support. Mostly it is based on our paper arxiv:1004.3631 and a talk given by the second author in the 2012 Abel symposium.