Fourier bases and a distance problem of Erd\H os

TL;DR

This paper proves that certain geometric sets cannot have non-harmonic exponential bases, using a combinatorial distance result by Erdős to establish the limitation.

Contribution

It introduces a new limitation on the existence of non-harmonic exponential bases in geometric sets, connecting harmonic analysis with combinatorial geometry.

Findings

No ball admits a non-harmonic orthogonal basis of exponentials.

Uses Erdős's distance problem to derive the main result.

Links harmonic analysis to combinatorial geometry constraints.

Abstract

We prove that no ball admits a non-harmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erd\H os, which says that the number of distances determined by $n$ points in ${\Bbb R}^d$ is at least $C_d n^{\frac{1}{d}+\epsilon_d}$, $\epsilon_d>0$.