Diophantine Approximation on Manifolds and Lower Bounds for Hausdorff Dimension

Given n ϵ ℕ and τ > 1/n, let S_nn.(τ) denote the classical set of τ-approximable points in ℝⁿ, which consists of x ϵ ℝⁿ that lie within distance q^-τ-1 from the lattice .(1/q)ℤⁿ for infinitely many q ϵ ℕ. In pioneering work, Kleinbock and Margulis showed that for any non-degenerate submanifold ℳ of ℝⁿ and any τ > 1/n almost all points on ℳ are not τ-Approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set ℳ ∩ S_n(τ). In this paper we suggest a new approach based on the Mass Transference Principle of Beresnevich and Velani [A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971-992], which enables us to find a sharp lower bound for dimℳ S_n(τ) for any C2 submanifold ℳ of ℝⁿ and any τ satisfying 1/n 6 ≥ τ 1/m. Here m is the codimension ofM.We also show that the condition on τ is best possible and extend the result to general approximating functions.