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

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.