Singular distributions, dimension of support, and symmetry of Fourier transform

TL;DR

This paper explores the relationship between the Fourier symmetry of measures on the circle and the size of their supports, extending Frostman's theorem and constructing critical examples with specific Fourier properties.

Contribution

It provides an extended version of Frostman's theorem linking Fourier decay and support dimension, and constructs compact supports supporting distributions with unique Fourier characteristics.

Findings

Extended Frostman's theorem relating Fourier decay to support dimension

Constructed compact supports for distributions with anti-analytic Fourier parts

Presented examples of measures with non-symmetry on small supports

Abstract

We study the "Fourier symmetry" of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are: (1) A one-side extension of Frostman's theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of the support; (2) A construction of compacts of "critical" size, which support distributions (even pseudo-functions) with anti-analytic part belonging to l^2. We also give examples of non-symmetry which may occur for measures with "small" support. A number of open questions are stated.