A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result:

If $\alpha$ is a real algebraic number and $\epsilon > 0$, then there exists only finitely many rational numbers $p/q$ with $q > 0$ and $(p,q) = 1$ such that $$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \epsilon}}$$ This result is famous for its vast improvement over previous results by Thue, Siegel, and Dyson and its ingenious proof, but is also notorious for being non-effective. That is, the result nor its (original) proof provides any insight as to how big the solutions (in $q$) can be, if any exists at all, or how many solutions there might be for a given $\alpha$ and $\epsilon$.

I have come to understand that to date no significant improvement over Roth's original proof has been made (according to my supervisor), and that the result is still non-effective. However, I am not so sure why it is so hard to make this result effective. Can anyone point to some serious attempts at making this result effective, or give a pithy explanation as to why it is so difficult?