On U- and M-sets for series with respect to characters of compact zero-dimensional groups

Generalizing our previous results on Walsh system, we consider the system of characters of zero-dimensional compact abelian group with the second axiom of countability and give for the sets of uniqueness with respect to this system a solution of category problem. We also obtain a generalization of a theorem on existence for this system a perfect M0M0-set whose Hausdorff h-measure equals zero.