5 research outputs found
Sphere tangencies, line incidences, and Lie's line-sphere correspondence
Two spheres with centers and and signed radii and are said to
be in contact if . Using Lie's line-sphere correspondence,
we show that if is a field in which is not a square, then there is an
isomorphism between the set of spheres in and the set of lines in a
suitably constructed Heisenberg group that is embedded in ; under
this isomorphism, contact between spheres translates to incidences between
lines.
In the past decade there has been significant progress in understanding the
incidence geometry of lines in three space. The contact-incidence isomorphism
allows us to translate statements about the incidence geometry of lines into
statements about the contact geometry of spheres. This leads to new bounds for
Erd\H{o}s' repeated distances problem in , and improved bounds for the
number of point-sphere incidences in three dimensions. These new bounds are
sharp for certain ranges of parameters.Comment: 20 pages, 2 figures. v2: minor changes in response to referee
comments. To appear in Math. Proc. Camb. Philos. So
Distinct distances in the complex plane
We prove that if is a set of points in , then either
the points in determine complex distances, or
is contained in a line with slope . If the latter occurs then each pair
of points in have complex distance 0.Comment: 41 pages, 0 figure