On the relationships between Fourier - Stieltjes coefficients and spectra of measures

TL;DR

This paper explores the connection between Fourier-Stieltjes coefficients and measure spectra, constructing specific examples of measures with particular spectral properties on the circle group.

Contribution

It introduces new examples of measures with natural spectra based on their Fourier coefficients, advancing understanding of spectral properties in harmonic analysis.

Findings

Constructed uncountable compact subsets with measures having natural spectra.

Identified an open set where measures with Fourier coefficients tending to zero also have natural spectra.

Enhanced the understanding of the relationship between Fourier coefficients and measure spectra.

Abstract

We construct examples of uncountable compact subsets of complex numbers with the property that any Borel measure on the circle group taking values of its Fourier coefficients from this set has natural spectrum. For measures with Fourier coefficients tending to 0 we construct tho open set with this property.