Size of orthogonal sets of exponentials for the disk

Abstract

Suppose $Λ\subseteq \RR^2$ has the property that any two exponentials with frequency from $Λ$ are orthogonal in the space $L^2(D)$, where $D \subseteq \RR^2$ is the unit disk. Such sets $Λ$ are known to be finite but it is not known if their size is uniformly bounded. We show that if there are two elements of $Λ$ which are distance $t$ apart then the size of $Λ$ is $O(t)$. As a consequence we improve a result of Iosevich and Jaming and show that $Λ$ has at most $O(R^{2/3})$ elements in any disk of radius $R$.