Mathematical Research Letters

Volume 6 (1999)

Number 2

Fourier bases and a distance problem of Erd\H os

Pages: 251 – 255

DOI: http://dx.doi.org/10.4310/MRL.1999.v6.n2.a13

Authors

Alex Iosevich (Georgetown University)

Nets Katz (University of Illinois at Chicago)

Steen Pedersen (Wright State University)

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 $C_d n^{\frac{1}{d}+\epsilon_d}$, $\epsilon_d >0$.

Full Text (PDF format)