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?
Best Answer
The non-effectivity, as far as I understand, is already present in Thue's Theorem, thus to understand it, one can look a the proof of the latter. The issue is roughly that, to show that there are not many "close rational approximations" $p/q$, one starts with the assumption that there exists one very close one $p_0/q_0$, and show that this very good approximation "repulses" or excludes other similarly or better ones. This of course doesn't work if the first $p_0/q_0$ doesn't exist... but such an assumption also gives the result! The ineffectivity is that we have no way of knowing which of the two alternatives has led to the conclusion.
There is a well-known analogy with the Siegel (or Landau-Siegel) zero question in the theory of Dirichlet $L$-functions. Siegel -- and it is certainly not coincidental that this is the same Siegel as in Thue--Siegel--Roth, though Landau did also have crucial ideas in that case -- proved an upper bound for real-zeros of quadratic Dirichlet $L$-functions by (1) showing that if there is one such $L$-function with a zero very close to $1$, then this "repulses" the zeros of all other quadratic Dirichet $L$-functions (this phenomenon is fairly well-understood under the name of Deuring-Heilbronn phenomenon), thus obtaining the desired bound; (2) arguing that if the "bad" $L$-function of (1) did not exist, then one is done anyway.
Here the ineffectivity is clear as day: the "bad" character of (1) is almost certainly non-existent, because it would violate rather badly the Generalized Riemann Hypothesis. But as far as we know today, we have to take into account the possibility of the existence of these bad characters... a possibility which however does have positive consequences, like Siegel's Theorem...
(There's much more to this second story; an entertaining account appeared in an article in the Notices of the AMS one or two years ago, written by J. Friedlander and H. Iwaniec.)