Covering theorems for uniqueness and extended uniqueness sets

Abstract

A covering theorem for a class of sets C asserts that every set in C can be covered by a countable union of sets in some (somehow simpler) class C. In the theory of sets of uniqueness on the unit circle T the first result of this kind is Piatetski-Shapiro's theorem in [PS], which states that every closed set of uniqueness can be covered by countably many closed sets in the class U'_1, consisting of those closed sets E ⊆ T for which there exists a sequence of functions in A(T), vanishing on E, which converges to the function 1 in the weak^*-topology.

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