On sets uniqueness for series in various systems of functions

It is proved that sets of first category are $\mathscr U$-sets for series in the Rademacher system. For series in the Faber–Schauder system with coefficients tending to zero it is proved that every countable set and every set of Cantor type with ratio $2^{-m}$ $(m=2,3,4,\dots)$ is a set of uniqueness.