An Inhomogeneous Jarník type theorem for planar curves
An Inhomogeneous Jarník type theorem for planar curves
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DOI：
10.1017/s0305004116000712
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发表时间：
2015-03
影响因子：
0.8
通讯作者：
D. Badziahin;Stephen Harrap;Mumtaz Hussain
中科院分区：
文献类型：
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作者：
D. Badziahin;Stephen Harrap;Mumtaz Hussain
在度量丢番图近似中，经典的近似主要有四类：齐次和非齐次情况下的同时近似和对偶近似。众所周知的Khintchine和Jarník的测度理论定理是它们的基础。最近，对于上述每一类的流形，在建立丢番图近似的度量理论方面已经取得了实质性的进展。特别地，除了一种情况外，Khintchine和Jarník-type的结果都适用于平面曲线的近似。本文证明了对偶逼近下平面曲线上收敛的一个非齐次Jarník型定理，从而完善了平面曲线上丢番图逼近的度量理论。
Abstract In metric Diophantine approximation there are classically four main classes of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental to each of them. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximation on manifolds for each of the classes above. In particular, both Khintchine and Jarník-type results have been established for approximation on planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves in the setting of dual approximation and in so doing complete the metric theory of Diophantine approximation on planar curves.