Some manifolds of Khinchin type for convergence
Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 175-193.

Récemment, Beresnevich, Vaughan, Velani et Zorin [2] ont donné des conditions suffisantes pour qu’une variété soit de type Khinchin pour la convergence. Nous montrons que leurs techniques peuvent être utilisées de manière plus optimale pour obtenir des résultats plus solides. Dans le processus, nous améliorons également un théorème de Dodson, Rynne et Vickers [5].

Recently, Beresnevich, Vaughan, Velani, and Zorin gave in [2] some sufficient conditions for a manifold to be of Khinchin type for convergence. We show that their techniques can be used in a more optimal way to yield stronger results. In the process we also improve a theorem of Dodson, Rynne, and Vickers [5].

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.1021
Classification : 11J13,  11J83,  11L07
Mots clés : Diophantine approximation, Khinchin type for convergence, Hausdorff dimension, rational points near manifolds
@article{JTNB_2018__30_1_175_0,
     author = {David Simmons},
     title = {Some manifolds of {Khinchin} type for convergence},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {175--193},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {1},
     year = {2018},
     doi = {10.5802/jtnb.1021},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1021/}
}
David Simmons. Some manifolds of Khinchin type for convergence. Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 175-193. doi : 10.5802/jtnb.1021. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1021/

[1] Victor Beresnevich; Robert Vaughan; Sanju Velani Inhomogeneous Diophantine approximation on planar curves, Math. Ann., Volume 349 (2011) no. 4, pp. 929-942 | Article | MR 2777039 | Zbl 1221.11158

[2] Victor Beresnevich; Robert Vaughan; Sanju Velani; Evgeniy Zorin Diophantine approximation on manifolds and the distribution of rationals: contributions to the convergence theory, Int. Math. Res. Not., Volume 2017 (2017) no. 10, pp. 2885-2908 | Article

[3] Vasilii Bernik; Maurice Dodson Metric Diophantine approximation on manifolds, Cambridge Tracts in Mathematics, Volume 137, Cambridge University Press, 1999, xi+172 pages | Zbl 0933.11040

[4] David A. Cox; John Little; Donal O’Shea Using algebraic geometry, Graduate Texts in Mathematics, Volume 185, Springer, 2005, xii+572 pages | MR 2122859 | Zbl 1079.13017

[5] Maurice Dodson; Bryan P. Rynne; J. A. G. Vickers Metric Diophantine approximation and Hausdorff dimension on manifolds, Math. Proc. Camb. Philos. Soc., Volume 105 (1989) no. 3, pp. 547-558 | Article | MR 985691 | Zbl 0677.10040

[6] Maurice Dodson; Bryan P. Rynne; J. A. G. Vickers Khintchine-type theorems on manifolds, Acta Arith., Volume 57 (1991) no. 2, pp. 115-130 | Article | MR 1092764 | Zbl 0736.11040

[7] Pertti Mattila Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, Volume 44, Cambridge University Press, 1995, xii+343 pages | Zbl 0819.28004

[8] Robert Vaughan; Sanju Velani Diophantine approximation on planar curves: the convergence theory, Invent. Math., Volume 166 (2006) no. 1, pp. 103-124 | Article | MR 2242634 | Zbl 1185.11047