  The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff
$f$-measure of the set of $\Psi$-approximable points on a nondegenerate
manifold. We refine and extend our previous work [Int. Math. Res. Not. IMRN
2021, no. 12, 8845--8867] in which we settled the problem (for dual
approximation) for hypersurfaces. We verify the GBSP for certain classes of
nondegenerate submanifolds of codimension greater than $1$. Concretely, for
codimension two or three, we provide examples of manifolds where the dependent
variables can be chosen as quadratic forms. Our method requires the manifold to
have even dimension at least the minimum of four and half the dimension of the
ambient space. We conjecture that these restrictions on the dimension of the
manifold are sufficient to provide similar examples in general.
