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?
Best Answer
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 $q_{k+1} \ll q_k^{7.617}$.