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Sphere tangencies, line incidences and Lie’s line-sphere correspondence

Part of: Discrete geometry

Published online by Cambridge University Press:  24 March 2021

JOSHUA ZAHL
JOSHUA ZAHL*
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada. e-mail: jzahl@math.ubc.ca
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Abstract

Two spheres with centers p and q and signed radii r and s are said to be in contact if |p–q|2=(r–s)2. Using Lie’s line-sphere correspondence, we show that if F is a field in which –1 is not a square, then there is an isomorphism between the set of spheres in F3 and the set of lines in a suitably constructed Heisenberg group that is embedded in (F[i])3; 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 Erdős’ repeated distances problem in F3, and improved bounds for the number of point-sphere incidences in three dimensions. These new bounds are sharp for certain ranges of parameters.

MSC classification

Primary: 52C10: Erdős problems and related topics of discrete geometry
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 2 , March 2022 , pp. 401 - 421
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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