On sets of orthogonal exponentials on the disk

Abstract

We show that if $A$ is a set of mutually orthogonal exponentials with respect to the unit disk then $|A\cap [-R,R{]}^2|{\lesssim }_{\epsilon }{R}^{3/5+\epsilon }$ holds. This improves the previous bound of $R^{2/3}$ by Iosevich--Kolountzakis. The main new ingredient in the proof is a discretized version of Marstrand's slicing theorem.

6 pages, improved presentation