Quasi-measures, Hausdorff $p$-measures and Walsh and Haar series

We study the classes of multiple Haar and Walsh series with at most polynomial growth of the rectangular partial sums. In terms of the Hausdorff $p$-measure, we find a sufficient condition (a criterion for the multiple Haar series) for a given set to be a $U$-set for series in the given class. We solve the recovery problem for the coefficients of the series in this class converging outside a uniqueness set. A Bari-type theorem is proved for the relative uniqueness sets for multiple Haar series. For one-dimensional Haar series, we get a criterion for a given set to be a $U$-set under certain assumptions that generalize the Arutyunyan–Talalyan conditions. We study the problem of describing those Cantor-type sets that are relative uniqueness sets for Haar series.