The Baker–Schmidt problem for dual approximation and some classes of manifolds

Hussain, Mumtaz; Schleischitz, Johannes

Comptes Rendus. Mathématique, Tome 362 (2024) no. G8, pp. 817-828

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.

Le problème de Baker–Schmidt généralisé (1970) concerne la mesure de Hausdorff $f$ de l’ensemble des points $\Psi$-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.

Classification : 28A78, 58C35
Keywords: Baker–Schmidt Problem, Hausdorff measure and dimension, Jarnik theorem
Mots-clés : Problème de Baker–Schmidt, mesure et dimension de Hausdorff, théorème de Jarnik

[1] Badziahin, Dzmitry; Beresnevich, Victor; Velani, Sanju Inhomogeneous theory of dual Diophantine approximation on manifolds, Adv. Math., Volume 232 (2013), pp. 1-35 | DOI | MR | Zbl

[2] Beresnevich, Victor A Groshev type theorem for convergence on manifolds, Acta Math. Hung., Volume 94 (2002) no. 1-2, pp. 99-130 | DOI | MR | Zbl

[3] Bernik, V. I. Application of the Hausdorff dimension in the theory of Diophantine approximations, Acta Arith., Volume 42 (1983) no. 3, pp. 219-253 | DOI | MR | Zbl

[4] Badziahin, Dzmitry; Harrap, Stephen; Hussain, Mumtaz An inhomogeneous Jarník type theorem for planar curves, Math. Proc. Camb. Philos. Soc., Volume 163 (2017) no. 1, pp. 47-70 | DOI | MR | Zbl

[5] Baker, Alan; Schmidt, Wolfgang M. Diophantine approximation and Hausdorff dimension, Proc. Lond. Math. Soc., Volume 21 (1970), pp. 1-11 | DOI | MR | Zbl

[6] Hussain, Mumtaz; Schleischitz, Johannes; Simmons, David Diophantine approximation on curves (2019) (https://arxiv.org/abs/1902.02094)

[7] Hussain, Mumtaz; Schleischitz, Johannes; Simmons, David The generalized Baker–Schmidt problem on hypersurfaces, Int. Math. Res. Not., Volume 2021 (2021) no. 12, pp. 8845-8867 | DOI | MR | Zbl

[8] Huang, Jing-Jing Hausdorff theory of dual approximation on planar curves, J. Reine Angew. Math., Volume 740 (2018), pp. 63-76 | DOI | MR | Zbl

[9] Huang, Jing-Jing The density of rational points near hypersurfaces, Duke Math. J., Volume 169 (2020) no. 11, pp. 2045-2077 | DOI | MR | Zbl

[10] Hussain, Mumtaz A Jarník type theorem for planar curves:everything about the parabola, Math. Proc. Camb. Philos. Soc., Volume 159 (2015) no. 1, pp. 47-60 | DOI | MR | Zbl

[11] Kleinbock, D. Y.; Margulis, G. A. Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. Math., Volume 148 (1998) no. 1, pp. 339-360 | DOI | MR | Zbl