Refinements of the holonomic approximation lemma

The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the $h-$principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working one pure partial derivative at a time is central to the discussion. We give applications of our results to flexible symplectic and contact topology.