The Baker-Schmidt problem for dual approximation and some classes of functions

TL;DR

This paper advances the understanding of the Generalised Baker-Schmidt Problem by extending results to certain nondegenerate submanifolds of higher codimension, especially those with quadratic forms, under specific dimensional conditions.

Contribution

It verifies the GBSP for classes of nondegenerate submanifolds of codimension greater than one, extending previous hypersurface results, with new examples involving quadratic forms.

Findings

Verified GBSP for certain nondegenerate submanifolds of codimension two and three.

Provided examples where dependent variables are quadratic forms.

Established dimensional conditions under which the results hold.

Abstract

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.