It will be shown that indicator functions mentioned in the
title exist 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. The results are new even for the real line end the unit circle.
(joint work with P. Perstneva)