  Given $n\in\mathbb{N}$ and $\tau>\frac1n$, let $\mathcal{S}_n(\tau)$ denote
the classical set of $\tau$-approximable points in $\mathbb{R}^n$, which
consists of ${\bf x}\in \mathbb{R}^n$ that lie within distance $q^{-\tau-1}$
from the lattice $\frac1q\mathbb{Z}^n$ for infinitely many $q\in\mathbb{N}$. In
pioneering work, Kleinbock $\&$ Margulis showed that for any non-degenerate
submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $\tau>\frac1n$ almost all
points on $\mathcal{M}$ are not $\tau$-approximable. Numerous subsequent papers
have been geared towards strengthening this result through investigating the
Hausdorff measure and dimension of the associated null set
$\mathcal{M}\cap\mathcal{S}_n(\tau)$. In this paper we suggest a new approach
based on the Mass Transference Principle, which enables us to find a sharp
lower bound for $\dim \mathcal{M}\cap\mathcal{S}_n(\tau)$ for any $C^2$
submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $\tau$ satisfying
$\frac1n\le\tau<\frac1m$. Here $m$ is the codimension of $\mathcal{M}$. We also
show that the condition on $\tau$ is best possible and extend the result to
general approximating functions.
