Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves
| dc.creator | Badziahin, Dzmitry | |
| dc.creator | Levesley, Jason | |
| dc.date | 2006-04-29 | |
| dc.date.accessioned | 2026-07-07T07:13:46Z | |
| dc.date.available | 2026-07-07T07:13:46Z | |
| dc.description | Let $\mathcal{C}$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation with two independent approximation functions; that is if a certain sum converges then the set of all points $(x,y)$ on the curve which satisfy simultaneously the inequalities $\| q x \| < ψ_1(q)$ and $\| qy \| < ψ_2(q)$ infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for the cae of multiplicative approximation $\| qx \| \| q y \| < ψ(q)$, we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup. | |
| dc.description | 13 pages | |
| dc.identifier | https://arxiv.org/abs/math/0605004 | |
| dc.identifier | http://arxiv.org/abs/math/0605004 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/112600 | |
| dc.subject | Number Theory | |
| dc.subject | 11J83 (Primary) 11J13, 11K60 (Secondary) | |
| dc.title | Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves | |
| dc.type | text |