Distance sets corresponding to convex bodies
| dc.creator | Kolountzakis, Mihail N. | |
| dc.date | 2003-03-17 | |
| dc.date.accessioned | 2026-07-07T04:56:08Z | |
| dc.date.available | 2026-07-07T04:56:08Z | |
| dc.description | 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 $ρ$. 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 $$ D_K(A) = \Set{\Norm{x-y}_K: x,y\in A} $$ 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 Łaba 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. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/math/0303212 | |
| dc.identifier | http://arxiv.org/abs/math/0303212 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66815 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Metric Geometry | |
| dc.title | Distance sets corresponding to convex bodies | |
| dc.type | text |