@article{4b98be2ef41b43339d178c0b5ad4b8fe,

title = "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.",

keywords = "Complement, Finite classical group, Non-degenerate subspace",

author = "Glasby, {S. P.} and Niemeyer, {Alice C.} and Praeger, {Cheryl E.}",

note = "Funding Information: We thank Jesse Lansdown for his help with computational investigations using the FinInG package [1] in GAP. The second and third authors thank the Hausdorff Research Institute for Mathematics, University of Bonn, for its hospitality during the International Workshop on Logic and Algorithms in Group Theory in 2018. We thank Gabriele Nebe for helpful discussions during the workshop. The authors gratefully acknowledge Australian Research Council Discovery Project Grant DP190100450. The second author acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project 286237555 - TRR 195. We are grateful to the two referees for their helpful suggestions. Funding Information: We thank Jesse Lansdown for his help with computational investigations using the FinInG package [1] in GAP . The second and third authors thank the Hausdorff Research Institute for Mathematics, University of Bonn, for its hospitality during the International Workshop on Logic and Algorithms in Group Theory in 2018. We thank Gabriele Nebe for helpful discussions during the workshop. The authors gratefully acknowledge Australian Research Council Discovery Project Grant DP190100450 . The second author acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project 286237555 - TRR 195 . We are grateful to the two referees for their helpful suggestions. Publisher Copyright: {\textcopyright} 2022",

year = "2022",

month = sep,

doi = "10.1016/j.ffa.2022.102055",

language = "English",

volume = "82",

journal = "Finite Fields and Their Applications",

issn = "1071-5797",

publisher = "Academic Press",

}