Diophantine approximation on planar curves and the distribution of rational points
| dc.creator | Beresnevich, Victor | |
| dc.creator | Dickinson, Detta | |
| dc.creator | Velani, Sanju | |
| dc.date | 2004-01-14 | |
| dc.date | 2006-04-26 | |
| dc.date.accessioned | 2026-07-07T06:35:55Z | |
| dc.date.available | 2026-07-07T06:35:55Z | |
| dc.description | Let $\cal C$ be a non--degenerate planar curve and for a real, positive decreasing function $ψ$ let $\cal C(ψ)$ denote the set of simultaneously $ψ$--approximable points lying on $\cal C$. We show that $\cal C$ is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on $\cal C$ of $\cal C(ψ)$ is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that $\cal C$ is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions $ψ$ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of $\cal C(ψ)$. These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds. | |
| dc.description | With an Appendix by Bob Vaughan: Sums of two squares near perfect squares | |
| dc.identifier | https://arxiv.org/abs/math/0401148 | |
| dc.identifier | http://arxiv.org/abs/math/0401148 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/99939 | |
| dc.subject | Number Theory | |
| dc.title | Diophantine approximation on planar curves and the distribution of rational points | |
| dc.type | text |