Intrinsic Diophantine approximation in Carnot groups and in the Siegel model of the Heisenberg group

TL;DR

This paper develops an intrinsic framework for Diophantine approximation on Carnot groups, especially the Heisenberg group, revealing new exponents, asymptotics, and connections to hyperbolic geometry and dynamical systems.

Contribution

It introduces a novel intrinsic Diophantine approximation theory on Carnot groups and distinguishes between Carnot and Siegel approximation in the Heisenberg group, with explicit exponents and geometric links.

Findings

Diophantine exponent for Carnot groups is (Q+1)/Q

Siegel Diophantine exponent in Heisenberg groups is 1

Siegel-badly approximable points are Schmidt winning

Abstract

We initiate the study of an intrinsic notion of Diophantine approximation on a rational Carnot group $G$. If $G$ has Hausdorff dimension $Q$, we show that its Diophantine exponent is equal to $(Q+1)/Q$, generalizing the case $G=\mathbb R^n$. We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group $\mathbb H^n$, distinguishing between two notions of Diophantine approximation by rational points in $\mathbb H^n$: Carnot Diophantine approximation and Siegel Diophantine approximation. After computing the Siegel Diophantine exponent (surprisingly, equal to 1 for all $\mathbb H^n$), we consider Siegel-badly approximable points to show that Siegel approximation is linked to both Heisenberg continued fractions and to geodesics in complex hyperbolic space. We conclude by showing that Carnot and Siegel approximation are qualitatively different: Siegel-badly approximable points are Schmidt winning in any complete Ahlfors regular subset of $\mathbb H^n$, while the set of Carnot-badly approximable points does not have this property.