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).



History question: Roth's theorem on approximating algebraic numbers...before Roth
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.


