Indicator functions with uniformly bounded Fourier sums and large gaps in the spectrum

TL;DR

This paper constructs indicator functions with uniformly bounded Fourier sums on locally compact Abelian groups, allowing large spectral gaps and small perturbations, and introduces a new Men$'$shov-type correction theorem.

Contribution

It develops a method to create indicator functions with bounded Fourier sums and large spectral gaps on arbitrary locally compact Abelian groups, including a weighted version and a correction theorem.

Findings

Constructed indicator functions with bounded Fourier sums on various groups

Achieved large gaps in the spectrum of these functions

Introduced a Men$'$shov-type correction theorem for Fourier analysis

Abstract

Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In the case of a noncompact group, the term "Fourier sums" should be understood as "partial Fourier integrals". A certain weighted version of the result is also provided. This version leads to a new Men$'$shov-type correction theorem.