Size of orthogonal sets of exponentials for the disk

Abstract

Suppose that $\Lambda \subseteq {\mathbb{R}}^2$ has the property that any two exponentials with frequency from $\Lambda$ are orthogonal in the space $L^2(D)$, where $D\subseteq {\mathbb{R}}^2$ is the unit disk. Such sets $\Lambda$ 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 $\Lambda$ which are distance $t$ apart then the size of $\Lambda$ is $O(t)$. As a consequence we improve a result of Iosevich and Jaming and show that $\Lambda$ has at most $O(R^{2/3})$ elements in any disk of radius $R$.