Convergence results in Diophantine approximation on manifolds

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.