  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.

Classical Analysis and ODEs