The probability of spanning a classical space by two non-degenerate subspaces of complementary dimensions

Abstract

Let n,n′ be positive integers and let V be an (n+n′)-dimensional vector space over a finite field F equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs (U,U′), where U is a non-degenerate n-subspace and U′ is a non-degenerate n′-subspace of V, such that U+U′=V (usually such spaces U and U′ are not perpendicular). The proportion is shown to be at least 1−c/|F| for some constant c<2 in the symplectic or unitary cases, and c<3 in the orthogonal case.