A Wiener-type theorem for arcs in the unit cirle

TL;DR

This paper establishes a Wiener-type theorem relating arc measures on the unit circle to Fourier coefficients, and applies it to analyze the Fourier series of the Cantor measure and its local dimension.

Contribution

It introduces a new Wiener-type theorem for arcs in the unit circle and applies it to measure analysis and fractal dimension computation.

Findings

Derived Fourier series for the Cantor measure

Computed the local dimension of specific measures

Established a new relation between arc measure and Fourier coefficients

Abstract

We prove a Wiener-type theorem for arcs in the unit circle which concerns express the measure of an arc in the unit circle via the measure's Fourier coefficients. Then we use it to give the Fourier series of the Cantor and to compute the local dimension of a measure satisfying certain conditions of Fourier coefficients.