Mathematical Research Letters

Volume 6, Issue 2, March 1999 pp. 251-255.

Fourier bases and a distance problem of Erd\H os

Authors: Alex Iosevich, Nets Katz, and Steen Pedersen
Author institution: Georgetown University, University of Illinois at Chicago, and Wright State University

Summary: 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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