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

TL;DR

This paper estimates the probability that two non-degenerate subspaces of complementary dimension in a finite-dimensional vector space over a finite field span the entire space, with bounds depending on the form type.

Contribution

It provides explicit lower bounds on the proportion of such pairs that span the whole space, extending understanding of subspace configurations over finite fields.

Findings

Proportion at least 1 - c/|F| for symplectic/unitary cases

Proportion at least 1 - c/|F| with c ≤ 2 or 3 for orthogonal case

Quantitative bounds on spanning probabilities in finite vector spaces

Abstract

Let $n,n'$ be positive integers and let $V$ be an $(n+n')$-dimensional vector space over a finite field $\mathbb{F}$ 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 $1-c/|\mathbb{F}|$ for some constant $c\leqslant 2$ in the symplectic or unitary cases, and $c<3$ in the orthogonal case.