Set of Uniqueness - Singular Distributions

Singular Distributions

A closed set is a set of uniqueness if and only if there exists a distribution S supported on the set (so in particular it must be singular) such that

( here are the Fourier coefficients). In all early examples of sets of uniqueness the distribution in question was in fact a measure. In 1954, though, Ilya Piatetski-Shapiro constructed an example of a set of uniqueness which does not support any measure with Fourier coefficients tending to zero. In other words, the generalization of distribution is necessary.