The Baker–Schmidt problem for dual approximation and some classes of manifolds
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) f...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-10-01
|
| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.585/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206178082193408 |
|---|---|
| author | Hussain, Mumtaz Schleischitz, Johannes |
| author_facet | Hussain, Mumtaz Schleischitz, Johannes |
| author_sort | Hussain, Mumtaz |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-aff42f78783e43aa956d6fb250d7e425 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-aff42f78783e43aa956d6fb250d7e4252025-02-07T11:22:49ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-10-01362G881782810.5802/crmath.58510.5802/crmath.585The Baker–Schmidt problem for dual approximation and some classes of manifoldsHussain, Mumtaz0Schleischitz, Johannes1Mumtaz Hussain, Department of Mathematical and Physical Sciences, La Trobe University, Bendigo 3552, AustraliaMiddle East Technical University, Northern Cyprus Campus, Kalkanli, Güzelyurt, CyprusThe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.585/Baker–Schmidt ProblemHausdorff measure and dimensionJarnik theorem |
| spellingShingle | Hussain, Mumtaz Schleischitz, Johannes The Baker–Schmidt problem for dual approximation and some classes of manifolds Comptes Rendus. Mathématique Baker–Schmidt Problem Hausdorff measure and dimension Jarnik theorem |
| title | The Baker–Schmidt problem for dual approximation and some classes of manifolds |
| title_full | The Baker–Schmidt problem for dual approximation and some classes of manifolds |
| title_fullStr | The Baker–Schmidt problem for dual approximation and some classes of manifolds |
| title_full_unstemmed | The Baker–Schmidt problem for dual approximation and some classes of manifolds |
| title_short | The Baker–Schmidt problem for dual approximation and some classes of manifolds |
| title_sort | baker schmidt problem for dual approximation and some classes of manifolds |
| topic | Baker–Schmidt Problem Hausdorff measure and dimension Jarnik theorem |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.585/ |
| work_keys_str_mv | AT hussainmumtaz thebakerschmidtproblemfordualapproximationandsomeclassesofmanifolds AT schleischitzjohannes thebakerschmidtproblemfordualapproximationandsomeclassesofmanifolds AT hussainmumtaz bakerschmidtproblemfordualapproximationandsomeclassesofmanifolds AT schleischitzjohannes bakerschmidtproblemfordualapproximationandsomeclassesofmanifolds |