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

Abstract

We study 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 $$\mathbf {H}^n$$, distinguishing between two notions of Diophantine approximation by rational points in $$\mathbf {H}^n$$: Carnot Diophantine approximation and Siegel Diophantine approximation. We provide a direct proof that the Siegel Diophantine exponent of $$\mathbf {H}^1$$ is equal to 1, confirming the general result of Hersonsky-Paulin, and then provide a link between Siegel Diophantine approximation, Heisenberg continued fractions, and geodesics in the Picard modular surface. 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 $$\mathbf {H}^n$$, while the set of Carnot-badly approximable points does not have this property.