Distance sets corresponding to convex bodies

Abstract

Suppose that $K \subseteq \RR^d$ is a 0-symmetric convex body which defines the usual norm $$ \Norm{x}_K = \sup\Set{t\ge 0: x \notin tK} $$ on $\RR^d$. Let also $A\subseteq\RR^d$ 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 $$ D_K(A) = \Set{\Norm{x-y}_K: x,y\in A} $$ contains all points $t\ge t_0$ for some positive number $t_0$. This was proved by Katznelson and Weiss, by Falconer and Marstrand and by Bourgain in the case where $K$ is the Euclidean ball in any dimension. As corollaries we obtain (a) an extension to any dimension of a theorem of Iosevich and \L aba 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.