Indicator Functions with Uniformly Bounded Fourier Sums and Large Gaps in the Spectrum

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 case a noncompact group, term “Fourier sums” should understood as “partial Fourier integrals”. A certain weighted version result is also provided. This leads to new Men $$'$$ shov-type correction theorem.