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

The gap between near commutativity and almost commutativity in symplectic category