Correction up to a function with sparse spectrum and uniformly convergent Fourier series

In 1984, the second author proved that, after correction on a set of arbitrarily small measure, any continuous function on a finite-dimensional compact Abelian group acquires sparse spectrum and uniformly convergent Fourier series. In the present paper we refine the result in two directions: first, we ensure uniform convergence in a stronger sense; second, we prove that the spectrum after correction can be put in even more peculiar sparse sets. Bibl. – 6 titles.