Complexity of Structure
The first evidence that sets of uniqueness have complex structure came from the study of Cantor-like sets. Salem and Zygmund showed that a Cantor-like set with dissection ratio ξ is a set of uniqueness if and only if 1/ξ is a Pisot number, that is an algebraic integer with the property that all its conjugates (if any) are smaller than 1. This was the first demonstration that the property of being a set of uniqueness has to do with arithmetic properties and not just some concept of size (Nina Bary had proved the case of ξ rational -- the Cantor-like set is a set of uniqueness if and only if 1/ξ is an integer -- a few years earlier).
Since the 50s, much work has gone into formalizing this complexity. The family of sets of uniqueness, considered as a set inside the space of compact sets (see Hausdorff distance), was located inside the analytical hierarchy. A crucial part in this research is played by the index of the set, which is an ordinal between 1 and ω1, first defined by Pyatetskii-Shapiro. Nowadays the research of sets of uniqueness is just as much a branch of descriptive set theory as it is of harmonic analysis. See the Kechris-Louveau book referenced below.
Read more about this topic: Set Of Uniqueness
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