Size of orthogonal sets of exponentials for the disk

TL;DR

This paper investigates the size limitations of orthogonal exponential sets for the disk, establishing bounds based on pairwise distances and improving previous results on their maximum size within a disk.

Contribution

The authors prove that the size of orthogonal exponential sets is linearly bounded by the distance between elements and provide an improved upper bound on their size in a disk.

Findings

Size of orthogonal exponential sets is O(t) when elements are t apart.

Maximum size of such sets in a disk of radius R is O(R^{2/3}).

Improves previous bounds on the size of orthogonal exponential sets.

Abstract

Suppose $\Lambda \subseteq \RR^2$ has the property that any two exponentials with frequency from $\Lambda$ are orthogonal in the space $L^2(D)$, where $D \subseteq \RR^2$ 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 $R$.