On the divergence of F\'ejer means with respect to Vilenkin systems on the set of measure zero

In the first part we review some crucial results concerning convergence/divergence of Fourier series and the F\'ejer means for various systems. As one new result we prove by concrete construction that for any set $E$ of measure zero there exists a function $f\in L_1(G_m)$ such that F\'ejer means with respect to Vilenkin systems diverge on this set. We use new constructions of Vilenkin polynomials, which was introduced in \cite{PTW2}.