Convergence results in Diophantine approximation on manifolds

描述：这个项目的目标是开发在公制丢番图近似中获得上界的技术。这一广泛的主题涉及例外丢番图集的测度论估计和建立Khintchine和Jarnik类型的结果。这个项目的具体目标将包括推广多项式的Jarnik-Besicovitch定理[1]的Hausdorff测度，并为平面曲线以外的欧氏空间的通用子流形和非齐次逼近发展类似的理论。特别地，我们将寻求推广黄[2]和Beresnevich[3]的工作。参考文献：[1]V.I.Bernik，Hausdorff维数在丢番图逼近理论中的应用。(俄语)Acta Arith。42(1983)，编号3,219-253。[2]J.-J.Huang，平面曲线上的对偶逼近的Hausdorff理论，将发表在Crelle的期刊上。[3]V.Beresnevich，流形上的一个Groshev型收敛定理。数学学报。亨加。94(2002)，1-2号，99-130号。

Description: The goal of this project to develop techniques for obtaining upper bounds in metric Diophantine approximation. This broad topic involves measure theoretic estimates for exceptional Diophantine sets and establishing Khintchine and Jarnik type results. The specific goals of this project will include a Hausdorff measure generalisation of the Jarnik-Besicovitch theorem for polynomials [1] and developing similar theory for generic submanifolds of a Euclidean space beyond planar curves and for inhomogeneous approximations. In particular, we will seek to extend the work of Huang [2] and Beresnevich [3].References:[1] V. I. Bernik, Application of the Hausdorff dimension in the theory of Diophantine approximations. (Russian) Acta Arith. 42 (1983), no. 3, 219-253. [2] J.-J. Huang, Hausdorff theory of dual approximation on planar curves, to appear in Crelle's Journal. [3] V. Beresnevich, A Groshev type theorem for convergence on manifolds. Acta Math. Hungar. 94 (2002), no. 1-2, 99-130.