On sets of orthogonal exponentials on the disk

TL;DR

This paper improves bounds on the size of sets of mutually orthogonal exponential functions on the disk, using a novel discretized slicing theorem to achieve tighter estimates.

Contribution

It introduces a new discretized Marstrand's slicing theorem to improve bounds on orthogonal exponential sets on the disk.

Findings

Bound |A ∩ [-R, R]^2| ≤ R^{3/5+ε} for orthogonal exponentials

Improves previous R^{2/3} bound by Iosevich--Kolountzakis

Uses a discretized version of Marstrand's slicing theorem

Abstract

We show that if $A$ is a set of mutually orthogonal exponentials with respect to the unit disk then $|A \cap [-R, R]^2| \lesssim_\varepsilon R^{3/5+\varepsilon}$ 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.