Equiangular lines and subspaces in Euclidean spaces

A family of lines through the origin in a Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in Rn was studied extensively for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, we prove that for every fixed angle θ and n sufficiently large, there are at most 2n − 2 lines in Rn with common angle θ. Moreover, this is achievable only for θ = arccos 1 3 . We also study analogous questions for k-dimensional subspaces. We discuss natural ways of defining the angle between k-dimensional subspaces and correspondingly study the maximum size of an equiangular set of k-dimensional subspaces in Rn, obtaining bounds which extend and improve a result of Blokhuis.