Correction up to functions with sparce spectrum and uniformly convergent Fourier integral representation: the group $\mathbb R^n$

This is an $\mathbb R^n$-conterpart of certain considerations on a similar subject for compact Abelian groups exposed by P. Ivanishvili and the author in 2010. The main difference with that paper is that certain notions and results of measure theory should be invoked in the case of $\mathbb R^n$.