The generalized Bari theorem for the Walsh system

For Walsh series in the Paley arrangement the author proves a generalized Bari theorem on the union of sets of uniqueness, from which it follows in particular that the union of two $\mathcal U$-sets, one of which is simultaneously an $F_\sigma$-set and a $G_\delta$-set, is a $\mathcal U$-set, and the union of two disjoint $\mathcal U$-sets of type $G_\delta$ is again a $\mathcal U$-set. It is shown that the last two assertions hold for sets of uniqueness of those classes of series for which the principle of localization of the kernel holds.