Abstract
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 U-sets with the G(p)-conditions introduced by W. Wade in 1981 is proved and a method for constructing such sets is given.
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