Size of orthogonal sets of exponentials for the disk

Abstract

Suppose Λ ⊆ R 2 has the property that any two exponentials with frequency from Λ are orthogonal in the space L 2 (D), where D ⊆ R 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.

Frame spectral pairs and exponential bases

“…There are many cases where it is known that a domain admits no orthogonal exponential bases. See, for example, [27,18,12,16] and the references contained therein. The conjecture is still open in dimensions d = 1, 2.…”