On a uniqueness theorem for functions with a sparse spectrum

Abstract

We present an example of a set {\( \wedge \in \mathbb{Z}\)} satisfying the following two conditions: (1) there exists a nonzero positive singular measure on the unit circle {\(\mathbb{T}\)} with spectrum in Λ; (2) if the spectrum of f∈L1 {\(\left( \mathbb{T} \right)\)} is contained in Λ and f vanishes on a set of positive measure, then f=0. Bibliography: 3 titles.