Diophantine approximation on matrices and Lie groups

TL;DR

This paper investigates extremality in metric Diophantine approximation on matrix submanifolds, providing criteria, explicit exponents, and applications to Lie groups, advancing understanding of approximation properties in algebraic and geometric contexts.

Contribution

It introduces a sharp criterion for extremality, links diophantine exponents to Zariski closures, and computes exponents for subgroups of nilpotent Lie groups.

Findings

The diophantine exponent depends only on the Zariski closure of the submanifold.

When the Zariski closure is rational, the exponent is rational and computable.

The method applies to random subgroups of nilpotent Lie groups, linking to representation theory.

Abstract

We study the general problem of extremality for metric Diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general, the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over the rational numbers, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest, in particular, we manage to determine the diophantine exponent of random subgroups of certain nilpotent Lie groups in terms of representation theoretic data.