The structure of the multidimensional Diophantine approximations

Let $l$ linear forms and $k$ quadratic forms $(n = l + 2k)$ be given in the $n$-dimensional real space $R$. Absolute values of the forms define a map of the space $R$ into the positive ortant $S_+$ of the $m$-dimensional real space $S$, where $m = l + k$. Here the integer lattice in $R$ is mapped into a set $\boldsymbol Z \subset S_+$. The closure of the convex hull $\boldsymbol G$ of the set $\boldsymbol Z\setminus 0$ is a polyhedral set. Integer points from $R$, which are mapped in the boundary $\partial\boldsymbol G$ of the polyhedron $\boldsymbol G$, give the best Diophantine approximations to root subspaces of all given forms. In the algebraic case, when the given forms are connected with roots of a polynomial of degree $n$, we prove that the polyhedron $\boldsymbol G$ has $m-1$ independent periods. It is a generalization of the Lagrange Theorem, that continued fractions of a square irrationality is periodic.