3:00 p.m., Friday (September 24, 2004)
Math Annex 1100
Department of Mathematics, UBC
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).