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

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=ℝ^n . We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group 𝐇^n , distinguishing between two notions of Diophantine approximation by rational points in 𝐇^n : Carnot Diophantine approximation and Siegel Diophantine approximation. We provide a direct proof that the Siegel Diophantine exponent of 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 𝐇^n , while the set of Carnot-badly approximable points does not have this property.