One can motivate Roth's theorem as follows. On $[0,1]$ consider the function $f$ that takes $x$ to the cardinality of the set
{ $p/q : |x - p/q | < 1/q^{2 + \epsilon}$ } .
Now one can see that $\int f dx <\infty$ by overestimating with a sum over q. This means that $f$ has finite values except on a set $E$ of measure 0. A random countable set would miss E, so one can hope that the irrational real algebraic numbers in $[0,1]$ will too.
But say now that we are only interested in denominators of the form $q=10^j$. The same heuristic strategy produces a conjecture much stronger than what Roth's Theorem offers. To wit, defining again a new function $f$ by the cardinalities of {$p/q, q = 10^j : |x - p/q | < \frac{1}{q \cdot (\ln q)^{1 + \epsilon}}$ } leads again to a function with a finite integral, and so the the prediction that algebraic numbers will have only finitely many approximations this good by numbers of the form $p/10^j$.
With so much room between Roth and and this prediction, my question is whether any result in the literature improves upon Roth in the direction of the prediction? More generally, are there known Roth variants where, at the expense of restricting the denominators $q$, one still has finiteness for a weaker standard of approximation?
