Fourier bases and a distance problem of Erd\H os

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 ${\mathbb{R}}^d$ is at least $C_d{n}^{\frac{1}{d}+{\epsilon }_d}$, ${\epsilon }_d>0$.