On sets of orthogonal exponentials on the disk

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