Fourier bases and a distance problem of Erd\H os

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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Published 1 January 1999