Counting basis extensions in a lattice

Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by $T$, producing an asymptotic estimate as $T \to \infty$. This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the $2$-dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in~$\mathbb R^n$. Our main result on counting basis extensions also generalizes to arbitrary lattices in~$\mathbb R^n$. Finally, we establish some basic properties of sparse representations of integers by multilinear forms.