Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin

Let T denote the unit circle in the plane. For various simple sets Λ in the plane we shall study the question whether (T,Λ) is a Heisenberg uniqueness pair. For example, we shall consider the cases where Λ is a circle or a union of two straight lines. We shall also use a theorem of Beurling and Malliavin.