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Roth's theorem has two universal quantifies, over irrational algebraic numbers $\alpha$ and over real $\epsilon>0$. Of course the theorem asserts in each instance that the inequality $$|\alpha-\frac{p}{q}|<\frac{1}{q^{2+\epsilon}}$$ has only finitely many solutions in integers $(p,q)$.

I seek confirmation (or not) for the following two historical claims:

Prior to Roth's paper...

1) the statement was not known to hold even for a single specific algebraic irrationality of degree $>2$ (but for all $\epsilon$);

2) the statement was not known to hold for single specific $\epsilon$ no matter how large (but for all $\alpha$).

The most famous precursors (Liouville--Thue--Siegel--Dyson) have $\epsilon$ depending on the degree of $\alpha$, so I feel pretty sure about 2), but less confident about 1).

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Just to point out a typo: the inequality should be $<1/q^{2+\epsilon}$ of course. I think (1) and (2) are correct too, but I don't know for certain. – Lucia Nov 22 '13 at 4:31
up vote 7 down vote accepted

I believe that both statements are correct. The ineffective results of Thue, Siegel, Gel'fond and Dyson do not lead to irrationality measures arbitrarily close to $2$, while earlier effective results (Thue, Siegel, etc) for classes of algebraic numbers give measures as close to $2$ as you like, but not of the $2 + \epsilon$ variety. If this last statement seems to be inconsistent, note that Thue proved irrationality measures for numbers like $\sqrt[3]{1+1/N}$, of the shape $2+c/\log N$ for some constant $c>0$. Even nowadays, we do not have an effective $2+\epsilon$ irrationality measure for even a single algebraic number of degree $3$ or higher.

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