Distance sets corresponding to convex bodies

Suppose that K ⊆ R, d ≥ 2, is a 0-symmetric convex body which defines the usual norm ‖x‖K = sup {t ≥ 0 : x / ∈ tK} on R. Let also A ⊆ R 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 DK(A) = {‖x− y‖K : x, y ∈ A} 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 à Laba regarding distance sets with respect to convex bodies of welldistributed 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.