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Consider the integer best approximations of a linear form in $n\ge 2$ realvariables. While it is well-known that any tail of this sequence always spans alattice is sharp for any $n\ge 2$. In this paper, we determine the exactHausdorff and packing dimension of the set where equality occurs, in terms of$n$. Moreover, independently we show that there exist real vectors whose bestapproximations 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 toan alternative constructive proof of Moshchevitin's result. We determine thepacking dimension and up to a small error term $O(n^{-1})$ also the Hausdorffdimension of the according set. Our method combines a new construction for alinear form in two variables $n=2$ with a result by Moshchevitin to amplifythem. We further employ the recent variatonal principle and some of itsconsequences, as well as estimates for Hausdorff and packing dimensions ofCartesian products and fibers. Our method permits much freedom for the inducedclassical exponents of approximation.

Metrical results on the geometry of best approximations for a linear form