A Cm Lusin approximation theorem for horizontal curves in the Heisenberg group

We prove a Cm Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is m- 1 times L1 differentiable almost everywhere coincides with a Cm horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if L1 differentiability is replaced by approximate differentiability. This shows our result is optimal and highlights differences between the Heisenberg and Euclidean settings.