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Fourier bases and a distance problem of Erd\H os

Alex Iosevich·Nets Katz·Steen Pedersen
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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 $C_d n^{\frac{1}{d}+ε_d}$, $ε_d>0$.

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Alex Iosevich, Nets Katz, Steen Pedersen. 2001-04-08. Fourier bases and a distance problem of Erd\H os. https://arxiv.org/abs/math/0104092

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