  This paper builds upon two key principles behind the Bourgain-Dyatlov
quantitative uniqueness theorem for functions with Fourier transform supported
in an Ahlfors regular set. We first provide a characterization of when a
quantitative uniqueness theorem holds for functions with very quickly decaying
Fourier transform, thereby providing an extension of the classical
Paneah-Logvinenko-Sereda theorem. Secondly, we derive a transference result
which converts a quantitative uniqueness theorem for functions with fast
decaying Fourier transform to one for functions with Fourier transform
supported on a fractal set. As well as recovering the result of
Bourgain-Dyatlov, we obtain analogous uniqueness results for denser fractals.
