The Monge problem of “piles and holes” on the torus and the problem of small denominators

We discuss the problem of existence of a smooth endomorphism of a closed $n$-dimensional manifold carrying a differential $n$-form into a prescribed volume form. Of course, we assume that the integrals of these forms over the whole manifold are equal. The solution of this problem for the $n$-dimensional torus reduces to the problem of small denominators well known in analysis.