The Baker–Schmidt problem for dual approximation and some classes of manifolds Le problème de Baker–Schmidt pour l’approximation duale et quelques classes de variétés

Hussain M., SCHLEISCHITZ J.

Comptes Rendus Mathematique, vol.362, pp.817-828, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 362
  • Publication Date: 2024
  • Doi Number: 10.5802/crmath.585
  • Journal Name: Comptes Rendus Mathematique
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aquatic Science & Fisheries Abstracts (ASFA), MathSciNet, zbMATH, Directory of Open Access Journals
  • Page Numbers: pp.817-828
  • Keywords: Baker–Schmidt Problem, Hausdorff measure and dimension, Jarnik theorem
  • Middle East Technical University Affiliated: Yes

Abstract

The Generalised Baker–Schmidt Problem (1970) concerns the Hausdorff f -measure of the set of Ψ-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.