  On any symplectic manifold of dimension greater than 2, we construct a pair
of smooth functions, such that on the one hand, the uniform norm of their
Poisson bracket equals to 1, but on the other hand, this pair cannot be
reasonably approximated (in the uniform norm) by a pair of Poisson commuting
smooth functions. This comes in contrast with the dimension 2 case, where by a
partial case of a result of Zapolsky, an opposite statement holds.
