[Paper Review] Metric Diophantine approximation for systems of linear forms via dynamics

Dmitry Kleinbock, G. A. Margulis+1 more|arXiv (Cornell University)|Apr 17, 2009

Mathematical Dynamics and Fractals23 references3 citations

TL;DR

This paper establishes the joint strong extremality of arbitrary finite collections of smooth nondegenerate submanifolds in $\mathbb{R}^n$ for systems of linear forms using dynamical methods. By proving generalized quantitative nondivergence estimates for translates of measures on the space of lattices, it extends the metric Diophantine approximation theory beyond single forms to dependent systems, showing that such measures are strongly extremal and thus almost every matrix in the submanifolds is not very well multiplicatively approximable.

ABSTRACT

The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong extremality' of arbitrary finite collection of smooth nondegenerate submanifolds of ${\bold R}^n$. The proofs are based on generalized quantitative nondivergence estimates for translates of measures on the space of lattices.

Motivation & Objective

• To generalize metric Diophantine approximation results for dependent quantities to systems of linear forms in $m \times n$ matrices.
• To establish that smooth nondegenerate submanifolds in $\mathbb{R}^n$ are jointly strongly extremal under the measure induced by their parameterization.
• To extend the framework of homogeneous dynamics to prove strong extremality for measures on $M_{m,n}$, including those arising from smooth maps from $\mathbb{R}^d$ to $M_{m,n}$.

Proposed method

• Use of generalized quantitative nondivergence estimates for translates of measures on the space of unimodular lattices in $\mathbb{R}^{m+n}$.
• Application of the theory of homogeneous flows to control the decay of measure of sets related to small values of linear forms.
• Leverage the dynamics of $\mathrm{SL}(m+n,\mathbb{R})$-actions on the homogeneous space $\mathrm{SL}(m+n,\mathbb{R}) / \mathrm{SL}(m+n,\mathbb{Z})$ to analyze approximation properties.
• Adapt techniques from [KM1] to the matrix setting by extending the notion of nondegeneracy to systems of forms.
• Prove that measures pushed forward by smooth nondegenerate maps from $\mathbb{R}^d$ to $M_{m,n}$ are strongly extremal via nondivergence estimates.
• Utilize the transference principle between homogeneous and inhomogeneous settings, showing that strong extremality implies inhomogeneous strong extremality under contracting measure conditions.

Experimental results

Research questions

• RQ1 Under what conditions on a smooth submanifold of $M_{m,n}$ is its Riemannian volume measure strongly extremal?
• RQ2 Can the strong extremality of measures associated with systems of linear forms be established using dynamical methods beyond the case $m=1$?
• RQ3 What is the precise relationship between the nondegeneracy of a map $F: \mathbb{R}^d \to M_{m,n}$ and the extremality of the pushforward measure $F_*\lambda$?
• RQ4 How do the concepts of very well approximable and very well multiplicatively approximable matrices behave under transposition in the matrix setting?
• RQ5 To what extent can Khintchine-type theorems be extended to smooth submanifolds of $M_{m,n}$ with $\min(m,n) > 1$?

Key findings

• The pushforward measure $F_*\lambda$ for any smooth nondegenerate map $F: \mathbb{R}^d \to M_{m,n}$ is strongly extremal, meaning $\mu$-almost every $Y \in M_{m,n}$ is not very well multiplicatively approximable.
• The joint strong extremality of any finite collection of smooth nondegenerate submanifolds of $\mathbb{R}^n$ is established, generalizing the single-form case.
• Generalized quantitative nondivergence estimates for measures on the space of lattices are proven and used as the key technical tool to control approximation rates.
• The measure $F_*\lambda$ is shown to be strongly contracting almost everywhere, which implies inhomogeneous strong extremality via results from [BV].
• The paper confirms that the set of very well multiplicatively approximable matrices has zero Lebesgue measure, and extends this to a broad class of smooth measures.
• The results imply that for any fixed $\mathbf{z} \in \mathbb{R}^m$, the set of matrices $Y$ for which $(Y,\mathbf{z})$ is very well multiplicatively approximable has zero measure under $F_*\lambda$.