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

Abstract

Abstract. In this article, we study an analytic curve ϕ : I = [a, b] → M(n × n, R) in the space of n by n real matrices, and show that if ϕ satisfies certain geometric conditions, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem is not improvable. To do this, we embed the curve into some homogeneous space G/Γ, and prove that under the action of some expanding diagonal flow A = {a(t) : t ∈ R}, the expanding curves tend to be equidistributed in G/Γ, as t → +∞. This solves a special case of a problem proposed by Nimish Shah in [12].

Equidistribution of expanding translates of curves and Diophantine approximation on matrices

“…Later Shah [Sha09b] studied the case m = 1, and showed that if an analytic curve in R n is not contained in any proper affine subspace, then almost every point on the curve is not DT-improvable. Lei Yang [Yan16a] studied the case m = n, and proved an analogous result for square matrices. These results have been generalized to supergeneric curves in M m×n (R) in the recent preprint [SY16], where an inductive algorithm was introduced to define generic and supergeneric curves.…”

Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

“…We use induction to complete the proof. For the case m = n, the lemma is due to Yang [20]. We provide a proof here.…”

“…Based on the same correspondence, Nimish Shah [18] proved the following stronger result: for m = 1 and general n, if an analytic curve ϕ : I = [a, b] → R n is not contained in a proper affine subspace, then almost every point on the curve is not DT -improvable. For m = n, Lei Yang [20] provided a geometric condition and proved that if an analytic curve ϕ : I = [a, b] → M(n × n, R) satisfies the condition, then almost every point on ϕ is not DT -improvable. The geometric condition given there provides some hint on solving the problem for general (m, n), and will be discussed in detail later.…”

“…3. In [20], it is proved that for m = n, if there exists s 0 ∈ I and a subinterval J s0 ⊂ I such that the derivative ϕ (s 0 ) is invertible, ϕ(s) − ϕ(s 0 ) is invertible for any s ∈ J s0 \ {s 0 }, and {(ϕ(s) − ϕ(s 0 )) −1 : s ∈ J s0 \ {s 0 }} is not contained in any proper affine subspace of M(m × m, R), then almost every point on the curve is not DT -improvable. Clearly this condition implies the supergeneric condition.…”

“…It turns out that this condition is the same as generic condition for m = 1. Later Yang [20] studied the case m = n.…”