Some manifolds of Khinchin type for convergence
David Simmons1
1 University of York Department of Mathematics Heslington, York YO10 5DD, UK
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].

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DOI : 10.5802/jtnb.1021
Classification : 11J13, 11J83, 11L07
Mots clés : Diophantine approximation, Khinchin type for convergence, Hausdorff dimension, rational points near manifolds
David Simmons 1

1 University of York Department of Mathematics Heslington, York YO10 5DD, UK
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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/

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[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 | DOI | MR | Zbl

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[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 | DOI | MR | Zbl

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

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

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