Mathematical Research Letters

Volume 6 (1999)

Number 2

Fourier bases and a distance problem of Erd\H os

Pages: 251 – 255



Alex Iosevich (Georgetown University)

Nets Katz (University of Illinois at Chicago)

Steen Pedersen (Wright State University)


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

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