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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?

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What about the converse? That is, for any fixed $r > 2$, consider the set $S$ of polynomials in $\mathbb{Z}[X]$ which have a real root $\alpha$ for which there exists a rational approximation $\vert\alpha -p/q\vert < q^{-r}$. Is it plausible that $S$ might not be computable? Maybe, even, it could be reduced to the Halting problem. Has this ever been considered? –  George Lowther Mar 18 '11 at 23:32
    
Alternatively, similar to the case of the rank of an elliptic curve and the Birch and Swinnerton-Dyer conjecture, are there any conjectures which are generally believed to be true and would lead to an effective Thue-Siegel-Roth theorem? –  George Lowther Mar 18 '11 at 23:47
    
Well, Felipe's response answers my question. An effective Thue-Siegel-Roth theorem is implied by the abc conjecture. –  George Lowther Mar 19 '11 at 2:49

5 Answers 5

up vote 17 down vote accepted

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.)

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Denis, are you referring to “What is … the parity phenomenon?” (ams.org/notices/200907/rtx090700817p.pdf)? –  L Spice Jul 3 '11 at 3:48

This is a kind of a late response, but the OP said "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 α and ϵ," and no one has addressed his comment about how many solutions. In fact, Roth's proof combined with a simple gap principle (and a lot of bookkeeping) gives a completely explicit upper bound for the number of solutions as a function of ϵ and the height of α. One can find versions of this in various places, including my paper: A quantitative version of Siegel's theorem: Integral points on elliptic curves and Catalan curves J. Reine Angew. Math. 378 (1987), 60-100. I have a vague recollection that Davenport may have been the first to point this out (maybe just for Thue or Siegel's theorem). There are also deep generalizations giving upper bounds for the number of exceptional subspaces in Schmidt's Subspace Theorem, see for example: A quantitative version of the absolute subspace theorem, J.-H. Evertse and H. P. Schlickewei, J. Reine Angew. Math. 548 (2002), 21–127.

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One way of looking at the issue is this: it is quite easy to transform the question of good rational approximations to algebraic numbers into a question about integral points on certain affine curves (Thue equations). Now the same issues come up: are there any integer points at all, are there only a finite number, and if so can you find them effectively? The finiteness here was taken care of by Siegel, when the genus is > 1. That was an ineffective proof also (sure, it came down to Siegel's version of TSR).

When Baker's method came along in the 1960s, some diophantine problems on curves of this general type became solvable, not just in theory (because Baker's methods are effective) but also in practice. Those were, however, examples where something special was working in our favour.

It is important to understand that while the proof method of Roth simply won't allow effectiveness, Baker's big advance was to remedy that defect by transcendence techniques. Baker's method has given some rather slim general effective results on TSR. Considering that Roth and Baker were both students of Davenport, I think it can be imagined that the techniques were looked over thoroughly. Decades have now gone by.

There isn't a slick answer here, I think. It is typical of tough research problems that we can say that a real advance will depend on a new idea, about which we don't have much clue right now.

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Others have mentioned the Baker-Feldman theorem and similar results coming from transcendence, which is the major source of weak effective general bounds in diophantine approximation. There is also the following paper, which deals with some special cases:

E. Bombieri, AJ van der Poorten, and JD Vaaler, Effective measures of irrationality for cubic extensions of number fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 211-248.

I don't think it is possible to adequately survey a big area of research in a MO post. If we knew why it is difficult to make it effective we might be able to prove something.

I should also add that both the ABC conjecture and Vojta's conjecture (which is a generalization of ABC, I guess) imply effective (perhaps up to some constant) versions of Roth's theorem, so we kind of know what to expect.

BTW this question is a duplicate of Question related to Diophantine approximations and Roth's theorem

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The techniques are often some sort of Padé approximation, or can be put in a framework of that. The work of the Chudnovskys had a linear diff equation in the background. This gives some cases that "work" I think, but I am no expert. springerlink.com/content/j02t25105r35g171 –  Junkie Mar 19 '11 at 4:44
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@Felipe: Do you mean that an effective ABC conjecture implies an effective version of Roth's theorem? If one only knows, say, that $\max(|a|,|b|,|c|) < K*N(abc)^2$, for some constant $K$, but no bound for $K$ is effectively known, can you deduce an effective version of Roth's theorem? One can certainly conceive of a Diophantine approximation style proof of the ABC conjecture that would lead to an ineffective constant $K$. –  Joe Silverman Jul 3 '11 at 2:42
    
@Joe: Yes, it all depends on what is meant by abc conjecture. See remarks 14.4.18 and 14.4.20 in the book of Bombieri and Gubler. –  Felipe Voloch Jul 3 '11 at 11:35
    
@Joe: And theorem 12.2.9 of op.cit. –  Felipe Voloch Jul 3 '11 at 11:40
    
@Felipe: Thanks for supplying the reference where Bombieri and Gubler say explicitly that one needs effective $ABC$ to get effective Roth. For those who don't have [BG], here's the exact quote from page 497: "Remark 14.4.20. The proof we have given actually shows that an effective version of the $K$-rational $abc$-conjecture implies effective versions of Roth's and Faltings's theorems." –  Joe Silverman Jul 4 '11 at 1:12

There has been progress towards effective Roth's theorem. Notably, Fel'dman was first to prove an effective power saving over Liouville's bound.

In the nutshell the source of ineffectivity comes from the following kind of argument. One obtains a sequence of positive real numbers $x_1,x_2,\dotsc,$ with the property that the product of any two distinct $x$'s is at most $1$. It immediately follows that the sequence is bounded, but of course this information does not yield any actual bound. In the Thue's proofs, one argues that no two rational approximation can be very good simultaneously, which is where such a sequence of $x$'s arises.

In my opinion, a good introduction to effective methods in transcendental number theory is in the notes by Waldschmidt.

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