Distance sets corresponding to convex bodies

Abstract

Suppose that $K \subseteq \mathbb{R}^{d}, d\ge 2$ is a 0-symmetric convex body which denes the usual norm

$$ \|x\|_{K} = \textrm{sup} \{t \ge 0:x \notin tK\} $$

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

$$ D_K(A) = \big \{\|{x-y}\|_K:x,y\in A \big\} $$

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.