  Consider the integer best approximations of a linear form in $n\ge 2$ real
variables. While it is well-known that any tail of this sequence always spans a
lattice is sharp for any $n\ge 2$. In this paper, we determine the exact
Hausdorff and packing dimension of the set where equality occurs, in terms of
$n$. Moreover, independently we show that there exist real vectors whose best
approximations lie in a union of two two-dimensional sublattices of $\Z^{n+1}$.
Our lattices jointly span a lattice of dimension three only, thereby leading to
an alternative constructive proof of Moshchevitin's result. We determine the
packing dimension and up to a small error term $O(n^{-1})$ also the Hausdorff
dimension of the according set. Our method combines a new construction for a
linear form in two variables $n=2$ with a result by Moshchevitin to amplify
them. We further employ the recent variatonal principle and some of its
consequences, as well as estimates for Hausdorff and packing dimensions of
Cartesian products and fibers. Our method permits much freedom for the induced
classical exponents of approximation.
