Diophantine approximation on matrices and Lie groups

Menny Aka,Lior Rosenzweig,Nicolas De Saxcé,Emmanuel Breuillard

doi:10.1007/s00039-018-0436-0

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 {mathbb{Q}}, 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 prove that the diophantine exponent of rational nilpotent Lie groups exists and is a rational number, which we determine explicitly in terms of representation theoretic data.

Highlights

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 Q, we prove that the exponent is rational and give a method to effectively compute it

Summary

A Zero-One Law

We turn to the problem of diophantine approximation in nilpotent Lie groups. The purpose of this section is to show the existence of a critical exponent for any connected nilpotent real Lie group (not necessarily defined over Q). Given β ≥ 0, the set of β-diophantine k-tuples in Gk is Lebesgue measurable and invariant under the action of Aut(Fk,s)(Q). Let G be a connected nilpotent Lie group equipped with a left-invariant distance d(·, ·), and Γ a finitely generated subgroup of G. Recall [ABRdS15a, Lemma 3.5.] that there is an integer M such that eMr(X1,...,Xk) belongs to the subgroup generated by eX1, . In view of Proposition 2.4, the set Dβ of k-tuples such that Γg is β-diophantine is measurable and invariant under the action of Aut(Fk,s)(Q). Since this action is ergodic by Proposition 2.2, we conclude that Dβ is either null or conull.

Critical Exponent for the Heisenberg Group

Diophantine Approximation and Flows on the Space of Lattices

The Submodularity Lemma

The Critical Exponent for Rational Nilpotent Lie Groups

Explicit Values of the Critical Exponent in Some Examples