I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ satisfying this inequality, one has $$ N(x) = \frac{\log(2x)\log(3x)}{2\log2\log3} + o\left(\frac{\log x}{\log\log x} \right). $$ The error term here depends critically on the diophantine nature of the real number $\frac{\log 2}{\log 3}$, and in particular on the growth of the denominators $q_m$ of its convergents (in the sense of continued fractions). One could obtain a power savings in $\log x$ above if one knew something like $q_{m+1} \ll q_m^A$ for some real number $A$, but I suspect proving such a bound is very hard. My knowledge of diophantine approximation is introductory at best, so I would like to know from any experts the following:

- Are there "standard" conjectures which predict the growth of $q_m$, at least in the case when one is approximating a quotient of logarithms of integers?
- Hardy proves that $q_{m+1} \leq e^{A q_m}$ for some explicit constant $A$, which he improves (using an old theorem of Pillai) to $e^{\varepsilon q_m}$ for any $\varepsilon > 0$ with $m > m_0(\varepsilon)$ . I suspect not much more is known in terms of a better bound unconditionally. Are there improvements to this estimate?