Geodesic
Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan)
Victor Beresnevich1 ; Detta Dickinson2 ; Sanju Velani3 ; R. C. Vaughan4
1 Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom
2 Mathematics Department, National University of Ireland, Maynooth, Co. Kildare, Ireland
3 Department of Mathematics, University of York, Heslington,York YO10 5DD, United Kingdom
4 Mathematics Department, Penn State University, State College, PA 16802, United States
Annals of mathematics, Tome 166 (2007) no. 2, pp. 367-426.

Voir la notice de l'article provenant de la source Annals of Mathematics website

Let $\mathcal{C}$ be a nondegenerate planar curve and for a real, positive decreasing function $\psi$ let $\mathcal{C}(\psi)$ denote the set of simultaneously $\psi$-approximable points lying on $\mathcal{C}$. We show that $\mathcal{C}$ is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on $\mathcal{C}$ of $\mathcal{C}(\psi)$ is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that $\mathcal{C}$ is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions $\psi$ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of $\mathcal{C}(\psi)$. These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.
DOI : 10.4007/annals.2007.166.367

Victor Beresnevich 1 ; Detta Dickinson 2 ; Sanju Velani 3 ; R. C. Vaughan 4

1 Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom
2 Mathematics Department, National University of Ireland, Maynooth, Co. Kildare, Ireland
3 Department of Mathematics, University of York, Heslington,York YO10 5DD, United Kingdom
4 Mathematics Department, Penn State University, State College, PA 16802, United States 
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Victor Beresnevich; Detta Dickinson; Sanju Velani; R. C. Vaughan. Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan). Annals of mathematics, Tome 166 (2007) no. 2, pp. 367-426. doi : 10.4007/annals.2007.166.367. http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.166.367/

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