The Generalised Baker–Schmidt Problem (1970) concerns the Hausdorff -measure of the set of -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 . 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.
Le problème de Baker–Schmidt généralisé (1970) concerne la mesure de Hausdorff de l’ensemble des points -approximables sur une variété non dégénérée. Nous affinons et étendons notre travail précédent [Int. Math. Res. Not. IMRN 2021, no. 12, 8845-8867] dans lequel nous avons résolu le problème (pour l’approximation duale) pour les hypersurfaces. Nous vérifions le GBSP pour certaines classes de sous-variétés non dégénérées de codimension supérieure à 1. Concrètement, pour la codimension deux ou trois, nous donnons des exemples de variétés où les variables dépendantes peuvent être choisies comme des formes quadratiques. Notre méthode exige que la variété ait une dimension paire au moins égale au minimum de quatre et à la moitié de la dimension de l’espace ambiant.
Nous conjecturons que ces restrictions sur la dimension de la variété sont suffisantes pour fournir des exemples similaires en général.
Revised:
Accepted:
Published online:
Keywords: Baker–Schmidt Problem, Hausdorff measure and dimension, Jarnik theorem
Mots-clés : Problème de Baker–Schmidt, mesure et dimension de Hausdorff, théorème de Jarnik
Mumtaz Hussain 1; Johannes Schleischitz 2

@article{CRMATH_2024__362_G8_817_0, author = {Mumtaz Hussain and Johannes Schleischitz}, title = {The {Baker{\textendash}Schmidt} problem for dual approximation and some classes of manifolds}, journal = {Comptes Rendus. Math\'ematique}, pages = {817--828}, publisher = {Acad\'emie des sciences, Paris}, volume = {362}, year = {2024}, doi = {10.5802/crmath.585}, language = {en}, }
TY - JOUR AU - Mumtaz Hussain AU - Johannes Schleischitz TI - The Baker–Schmidt problem for dual approximation and some classes of manifolds JO - Comptes Rendus. Mathématique PY - 2024 SP - 817 EP - 828 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.585 LA - en ID - CRMATH_2024__362_G8_817_0 ER -
Mumtaz Hussain; Johannes Schleischitz. The Baker–Schmidt problem for dual approximation and some classes of manifolds. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 817-828. doi : 10.5802/crmath.585. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.585/
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