Distance sets corresponding to non-Euclidean norms

If E is a subset of a normed space X, we define its distance set to be the set of all distances between pairs of points in E. We consider the following general question: if the size of E is given, what is the minimum size of its distance set? For finite sets in Euclidean spaces, this is a notoriously hard question due to Erdos, which remains open despite many efforts by top combinatorialists. The ``continuous" version, due to Falconer, also remains unsolved.

In this talk, we will discuss the recent work on variants of these problems for more general finite-dimensional normed spaces. Roughly, if the unit ball (in the new norm) is strictly convex, we are able to apply methods borrowed from the Euclidean case, though there are complications. On the other hand, ``polygonal" norms (e.g. \ell^\infty) are quite different and allow much smaller distance sets; we are in fact able to give an exact characterization of the extremal cases. Our methods range from elementary geometry to number theory to Fourier analysis.

Results presented in this talk were obtained jointly with Alex Iosevich and with Sergei Konyagin.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).