Talk:Roth's theorem

As I recall, there is some logical point about the effective nature of C(&epsilon;) that has been brought up (by Roquette?); such that it is recursive in a definite sense, though it is not going to be provably primitive recursive or anything very useful by the method. Charles Matthews 10:20, 6 Jul 2004 (UTC)

I think it should be mentioned somewhere that &alpha; is irrational. The second inequality is not true if &alpha; is rational, since then |&alpha; - p/q| may be zero (I'm assuming that C(&epsilon;) should be positive). I don't know about the first inequality though, which is why I didn't edit the article myself. -- Jitse Niesen 10:15, 2 Nov 2004 (UTC)

Yes, good point. Actually the first inequality should be OK for rational alpha, since rationals are far enough from other rationals. The statement of the second does need to say non-zero. Charles Matthews 10:47, 2 Nov 2004 (UTC)

What is the connection to Liouville's theorem? This was not clear to me from the article. Gadykozma 13:47, 2 Nov 2004 (UTC)

It is the case of exponent d, where d is the degree. This is too weak to have implications for diophantine equations (basically because it is proved by saying there is no integer n with 0 < n < 1). Thue first sharpened the exponent, winning a factor of 2. Roth's result is the best possible exponent.

Charles Matthews 16:02, 2 Nov 2004 (UTC)

My guess is that the Liouville's theorem mentioned in the article is neither Liouville's theorem (complex analysis) nor Liouville's theorem (Hamiltonian) but another theorem about a lower bound for the distance between roots of a polynomial with integer coefficients. Unfortunately, I never studied this part of maths properly. -- Jitse Niesen 16:39, 2 Nov 2004 (UTC)

Oh, sorry, that Liouville's theorem is what is given as a lemma at Liouville number. Obviously there needs to be some work done about that.

Charles Matthews 17:04, 2 Nov 2004 (UTC)

I gave it a try; please check and improve. -- Jitse Niesen 17:38, 2 Nov 2004 (UTC)

Dubious
This seems wrong:


 * Later work using the methods of Alan Baker made some small impact on effective improvements to Liouville's theorem on diophantine approximation, which gives a bound


 * $$\left|\alpha-{p \over q}\right| \geq \frac{C}{q^d}$$


 * (see Liouville number); but the inequalities are still weak.

Liouville's theorem is effective already, and Baker's work made more than a small impact. Probably best to cite Hindry and Silverman, ISBN 0387989811, here. Thue Siegel Roth (talk) 14:23, 8 February 2012 (UTC)