Distance sets corresponding to convex bodies

Suppose that K subset of or equal to R-d, d greater than or equal to 2, is a 0-symmetric convex body which denes the usual norm parallel toxparallel to(K) = sup {t greater than or equal to 0:x is not an element of tK} on R-d. Let also A subset of or equal to R-d be a measurable set of positive upper density rho. We show that if the body K is not a polytope, or if it is a polytope with many faces (depending on rho), then the distance set D-K(A) = {parallel tox-yparallel to(K):x,y is an element of A} contains all points t greater than or equal to t(0) for some positive number t(0) . 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.