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 $$(Q+1)/Q$$ ( Q + 1 ) / Q , generalizing the case $$G=\mathbb {R}^n$$ 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 $$\mathbf {H}^n$$ H n , distinguishing between two notions of Diophantine approximation by rational points in $$\mathbf {H}^n$$ H n : Carnot Diophantine approximation and Siegel Diophantine approximation. We provide a direct proof that the Siegel Diophantine exponent of $$\mathbf {H}^1$$ 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 $$\mathbf {H}^n$$ H n , while the set of Carnot-badly approximable points does not have this property.

中文翻译：

---

卡诺群和海森堡群的 Siegel 模型中的内在丢番图近似

我们研究了有理卡诺群 G 上丢番图近似的内在概念。如果 G 有 Hausdorff 维 Q ，我们证明它的丢番图指数等于 $$(Q+1)/Q$$ ( Q + 1 ) / Q ，概括情况 $$G=\mathbb {R}^n$ $ G = R n 。此外，我们还获得了有理近似计数的精确渐近线。然后，我们关注海森堡群 $$\mathbf {H}^n$$ H n 的情况，通过 $$\mathbf {H}^n$$ H n 中的有理点区分丢番图逼近的两个概念：卡诺Diophantine 近似和 Siegel Diophantine 近似。我们提供了 $$\mathbf {H}^1$$ H 1 的 Siegel Diophantine 指数等于 1 的直接证明，证实了 Hersonsky-Paulin 的一般结果，然后提供了 Siegel Diophantine 近似之间的联系，海森堡继续分数，和皮卡德模块化表面中的测地线。我们通过证明 Carnot 和 Siegel 近似在性质上是不同的得出结论：Siegel-badly approximate point is Schmidt 在 $$\mathbf {H}^n$$H n 的任何完全 Ahlfors 正则子集中获胜，而 Carnot-badly 近似的集合点没有这个属性。