Spectral gaps for sets and measures

TL;DR

This paper investigates the spectral gaps of measures supported on closed subsets of the real line, aiming to relate the maximum spectral gap to the metric properties of the support set.

Contribution

It introduces a method to estimate the spectral gap size based on the geometric and metric characteristics of the support set.

Findings

Derived bounds for spectral gaps in terms of set properties

Established relationships between support set metrics and spectral spectrum

Provided new insights into spectral analysis of measures on real sets

Abstract

If $X$ is a closed subset of the real line, denote by $\GG_X$ the supremum of the size of the gap in the Fourier spectrum, taken over all non-trivial finite complex measures supported on $X$. In this paper we attempt to find $\GG_X$ in terms of metric properties of $X$.