Editing Set of uniqueness (section)

== Complexity of structure ==
The first evidence that sets of uniqueness have complex structure came from the study of [[Cantor set|Cantor-like sets]]. [[Raphaël Salem]] and [[Antoni Zygmund|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–Vijayaraghavan number|Pisot number]], that is an [[algebraic integer]] with the property that all its [[Conjugate element (field theory)|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 Bari]] 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{{Clarify|date=March 2023|reason=1950s?}}, 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 number|ordinal]] between 1 and ω<sub>1</sub>, 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.