# The growth of certain continued fractions

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:

1. 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?
2. 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?

The keyword you are looking for is "irrationality measure" -- I think some authors (such as Lang) call it constant type. If you know the irrationality measure of $$\alpha$$ is $$\mu = \mu(\alpha)$$, then the convergents of $$\alpha$$ satisfy $$q_{k+1} \ll q_k^{\mu -1 + \epsilon}$$ for every $$\epsilon>0$$.
Most irrational numbers you can write down will have irrationality measure $$2$$. This can be made precise in the sense that a standard probabilistic argument shows that the set of $$\alpha$$ with $$\mu(\alpha)>2$$ has null Lebesgue measure. Showing this for explicit numbers, however, can be very hard -- for example, Roth's theorem (that famously probably won him a Fields medal) states that $$\mu(\alpha) = 2$$ if $$\alpha$$ is an algebraic irrational.
I'm not sure what is best known for the irrationality measures of quotients of logarithms, but I found this previous MO question when looking it up. Lucia seems to think that's the state of the art (at least circa 2014), and that gives something like $$q_{k+1} \ll q_k^{7.617}$$.
• The original application I had in mind was to numbers $x < m^a n^b \leq c x$ with $m,n$ fixed and $a,b$ lying in prescribed residue classes, and $c$ constant or varying perhaps slightly with $x$. This requires a sufficiently good error term in the (generalized) counting problem I mentioned if you want to say something for all $x$ and not just sufficiently large $x$. Dec 13, 2022 at 21:30