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)