On metric diophantine approximation in matrices and Lie groups

Equidistribution of expanding curves in homogeneous spaces and Diophantine approximation on square matrices

“…Kleinbock, Margulis and Wang [6] later gave a necessary and sufficient condition of a submanifold of M(m×n, R) being extremal. Recently, Aka, Breuillard, Rosenzweig and de Saxcé [1] gave a family of subvarieties of M(m × n, R), and announced a theorem stating that if a submanifold U ⊂ M(m × n, R) is not contained in any one of the subvarieties given above, then U is extremal. It turns out that condition A.2 in Theorem 1.2 is stronger than the condition given in [1].…”

“…Recently, Aka, Breuillard, Rosenzweig and de Saxcé [1] gave a family of subvarieties of M(m × n, R), and announced a theorem stating that if a submanifold U ⊂ M(m × n, R) is not contained in any one of the subvarieties given above, then U is extremal. It turns out that condition A.2 in Theorem 1.2 is stronger than the condition given in [1]. We will discuss it in detail in Appendix A.…”

“…Acknowledgement: I would like to express my deep gratitude to my advisor, Professor Nimish Shah, for suggesting this problem to me, and his continuous advise and support during the process of the work. I also would like to thank Professor Kleinbock for reading an earlier version of this article and giving me a lot of comments and suggestions, for example, drawing [6] and [1] to my attention and suggesting me to figure out the relation between the geometric conditions given in this article and the arithmetic condition given in [1].…”

“…1. Since ϕ is analytic, if its derivative ϕ (1) (s) is invertible at some point, then it must be invertible at every point except finite many points.…”

“…2. If ϕ satisfies condition A.1 above, then for every s 0 ∈ I such that ϕ (1) (s 0 ) is invertible, there always exists some subinterval J s0 of I, such that ϕ(s) − ϕ(s 0 ) is invertible for all s ∈ J s0 . This is because when s is close to s 0 , the major part of ϕ(s…”