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?

1 Answer 1


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

  • $\begingroup$ Thanks for the answer. I knew about irrationality measure, but I didn't think to phrase it that way in terms of convergents. $\endgroup$ Dec 13, 2022 at 8:32
  • $\begingroup$ @JoshuaStucky: Sure! Do you mind sharing a reference for Hardy's result? It seems pretty cool. $\endgroup$ Dec 13, 2022 at 9:30
  • 1
    $\begingroup$ It's Chapter 5 of his book "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work" $\endgroup$ Dec 13, 2022 at 21:25
  • $\begingroup$ 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 $x$. $\endgroup$ Dec 13, 2022 at 21:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.