Uniqueness Questions for Some Classes of Haar Series

Sets of relative uniqueness for Haar series are studied. Whole classes of conditions on the behavior of a Haar series, including the Arutyunyan–Talalyan condition, are considered. New numerical characteristic of perfect sets are introduced. They are used to obtain necessary conditions and sufficient conditions for a given set to be a set of relative uniqueness under certain assumptions. Thereby, the 1967 results of G. M. Mushegyan are generalized. Moreover, for $0<p<2$, the existence of perfect $\operatorname{U}$-sets with the $\operatorname{G}(p)$-conditions introduced by W. Wade in 1981 is proved and a method for constructing such sets is given.