  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.
