Abstract
Suppose that \( K \subseteq \mathbb{R}^{d}, d\ge 2 \) is a 0-symmetric convex body which denes the usual norm
on \( \mathbb{R}^{d} \) . Let also \( A \subseteq \mathbb{R}^{d} \) be a measurable set of positive upper density ρ. We show that if the body K is not a polytope, or if it is a polytope with many faces (depending on ρ), then the distance set
contains all points t ≥ t0 for some positive number t0 . This was proved by Furstenberg, Katznelson and Weiss, by Falconer and Marstrand and by Bourgain in the case where K is the Euclidean ball in any dimension greater than 1. As corollaries we obtain (a) an extension to any dimension of a theorem of Iosevich and Laba regarding distance sets with respect to convex bodies of well-distributed sets in the plane, and also (b) a new proof of a theorem of Iosevich, Katz and Tao about the nonexistence of Fourier spectra for smooth convex bodies with positive curvature.
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Kolountzakis, M. Distance sets corresponding to convex bodies. Geom. funct. anal. 14, 734–744 (2004). https://doi.org/10.1007/s00039-004-0472-9
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DOI: https://doi.org/10.1007/s00039-004-0472-9